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Pinnacle Day: 89th Coordinate Geometry
Coordinate Geometry
Coordinate geometry is a system of
geometry where the position of points on
the plane is described by using an
ordered pair of numbers. The x-
coordinate and the y- coordinate of a
point taken together is known as
coordinates of the point
Key points:
Coordinate Axes : The perpendicular •
lines in the cartesian system are x axis
and y axis . Both of them together are
known as coordinate axes.
Origin : The point of intersection of •
coordinate axes is origin. The
coordinates of origin are (0 , 0).
Abscissa : The x - coordinate is called •
abscissa. It is the perpendicular distance
of a point from y axis.
Ordinate : The y-coordinate is called •
ordinate . It is the perpendicular distance
of a point from x-axis.
If y = a, where a is constant then y = a •
denotes a straight line parallel to x-axis.
If x = a, where a is constant then x = a •
denotes a straight line parallel to y-axis.
For passing a line through the origin, •
one of the coordinates of the given
equation must be (0, 0).
Important Formulae :
Distance Formula : If points P(x
1
, y
1
), •
Q(x
2 ,
y
2
) lie on the plane xy then,
PQ = ( ??
2
- ??
1
)
2
+ ( ??
2
- ??
1
)
2
Section Formula : •
The coordinates of a point which divides
the line segment joined by two distinct
points (x
1
, y
1
) and (x
2
, y
2
) in the ratio
m : n are :
(x , y) = ( ,
?? ??
2
+ ?? ??
1
?? + ??
?? ??
2
+ ?? ??
1
?? + ??
)
Note : When line segment is divided
externally in the ratio m : n then ratio
becomes ( ,
?? ??
2
- ?? ??
1
?? - ??
?? ??
2
- ?? ??
1
?? - ??
)
Midpoint Formula : •
It is derived from the Section Formula.
The coordinates of a point which divides
the line segments in the ratio 1 : 1. Then,
(x, y) = ( , )
??
1
+ ??
2
2
??
1
+ ??
2
2
Centroid Formula : •
If (x
1
, y
1
) , (x
2
, y
2
) and (x
3
, y
3
) are the
vertices of a triangle then coordinates of
the centroid are :
( , )
??
1
+ ??
2
+ ??
3
3
??
1
+ ??
2
+ ??
3
3
Incentre Formula : •
If (x
1
, y
1
) , (x
2
, y
2
) and (x
3
, y
3
) are the
vertices of a triangle then coordinates of
the incenter are :
( , )
?? ??
1
+ ?? ??
2
+ ?? ??
3
?? + ?? + ??
?? ??
1
+ ?? ??
2
+ ?? ??
3
?? + ?? + ??
Slope of a line : •
If (x
1 ,
y
1
) , (x
2
, y
2
) are any two points on
line L, then the ratio is called the
??
2
- ??
1
??
2
- ??
1
slope of the line L.
Angle between two lines of slope m
1
•
and m
2,
respectively then
tan = ?
??
2
- ??
1
1 + ??
1
??
2
|
|
|
|
|
|
Intercept form of equation of a line : •
+ = 1,where ‘a’ is x intercept of line
??
??
??
??
and ‘b’ is y intercept of line .
Perpendicular distance from a point (x
1
, •
y
1
) to line ax + by + c = 0, then
D =
?? ??
1
+ ?? ??
1
+ ??
| |
??
2
+ ??
2
Distance between lines : • ?
a
1
x + b
1
y + c
1
= 0 and a
2
x + b
2
y + c
2
= 0 is
D =
??
1
- ??
2
??
2
+ ??
2
|
|
|
|
|
|
Equation of a circle •
+ + 2gx + 2fy + c = 0, where center ??
2
??
2
of the circle is (- g , - f) and radius =
??
2
+ ??
2
- ??
If point A (x
1
, y
1
) , B(x
2
, y
2
) and C (x
3
, y
3
) •
are the vertices of a triangle then area of
a triangle is :
1
2
??
1
( ??
2
- ??
3
) + ??
2
( ??
3
- ??
1
) + ??
3
( ??
1
- ??
2
)
| |
Equation of a circle - + - = ? ( ?? h )
2
( ?? ?? )
2
??
2
Where
( ) x, y coordinate of center point h , ?? ?
x x - coordinate of circle point ?
y y - coordinate of circle point ?
Variety Questions
Q.1. What is the area (in unit squares) of
the triangle enclosed by the graphs of
the equations 2x + 5y = 12, x + y = 3 and
the x-axis?
SSC CGL Tier II (03/02/2022)
(a) 2.5 (b) 3.5 (c) 3 (d) 4
Q.2. The equation of circle with centre (1
, – 2) and radius 4 cm is :
SSC CHSL 17/03/2020 (Afternoon)
(a) ??
2
+ ??
2
+ 2 ?? - 4 ?? = 16
(b) ??
2
+ ??
2
- 2 ?? + 4 ?? = 16
(c) ??
2
+ ??
2
+ 2 ?? - 4 ?? = 11
(d) ??
2
+ ??
2
- 2 ?? + 4 ?? = 11
Q.3. What is the area (in square units) of
the triangular region enclosed by the
graphs of the equations x + y = 3,
2x + 5y = 12 and the x axis?
SSC CGL Tier II (13/09/2019)
(a) 2 (b) 3 (c) 4 (d) 6
Practice Questions
SSC CHSL 2023 Tier - 1
Q.4. Find the radius of the circle
. ?? ² + ?? ² = 25
SSC CHSL 08/08/2023 (4th Shift)
(a) 25 units (b) 5 units
(c) 2 units (d) 12 units
SSC CGL 2022 Tier - 2
Q.5. Find the coordinates of the points
where the graph 57x 19y = 399 cuts -
the coordinate axis.
SSC CGL Tier II (07/03/2023)
(a) x-axis at (-7,0) and y-axis at (0,-21)
(b) x-axis at (-7,0) and y-axis at (0,21)
(c) x-axis at (7,0) and y-axis at (0,-21)
(d) x-axis at (7,0) and y-axis at (0,21)
SSC CGL 2021 Tier - 2
Q.6. The graph of the equation x = a
(a 0) is a ______. ?
SSC CGL Tier II (08/08/2022)
(a) line parallel to x axis
(b) line parallel to y axis
(c) line at an angle of 45 degree to y axis
(d) line at an angle of 45 degree to x axis
SSC CGL 2020 Tier - 2
Q.7. The graphs of the equations
4x + y = and x + y + = 0
1
3
8
3
1
2
3
4
5
2
and intersect at a point P . The point P
also lies on the graph of the equation:
SSC CGL Tier II (29/01/2022)
(a) 4x y 7 0 (b) x 3y 12 0 - + = - - =
(c) 3x y 7 0 (d) x 2y 0 - - = + - 5 =
Q.8. What is the area (in unit squares) of
the region enclosed by the graphs of the
equations 2x – 3y + 6 = 0, 4x + y = 16 and
y = 0 ?
SSC CGL Tier II (29/01/2022)
(a) 11.5 (b) 14 (c) 10.5 (d) 12
Q.9. The graphs of the equations
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Page 2
Pinnacle Day: 89th Coordinate Geometry
Coordinate Geometry
Coordinate geometry is a system of
geometry where the position of points on
the plane is described by using an
ordered pair of numbers. The x-
coordinate and the y- coordinate of a
point taken together is known as
coordinates of the point
Key points:
Coordinate Axes : The perpendicular •
lines in the cartesian system are x axis
and y axis . Both of them together are
known as coordinate axes.
Origin : The point of intersection of •
coordinate axes is origin. The
coordinates of origin are (0 , 0).
Abscissa : The x - coordinate is called •
abscissa. It is the perpendicular distance
of a point from y axis.
Ordinate : The y-coordinate is called •
ordinate . It is the perpendicular distance
of a point from x-axis.
If y = a, where a is constant then y = a •
denotes a straight line parallel to x-axis.
If x = a, where a is constant then x = a •
denotes a straight line parallel to y-axis.
For passing a line through the origin, •
one of the coordinates of the given
equation must be (0, 0).
Important Formulae :
Distance Formula : If points P(x
1
, y
1
), •
Q(x
2 ,
y
2
) lie on the plane xy then,
PQ = ( ??
2
- ??
1
)
2
+ ( ??
2
- ??
1
)
2
Section Formula : •
The coordinates of a point which divides
the line segment joined by two distinct
points (x
1
, y
1
) and (x
2
, y
2
) in the ratio
m : n are :
(x , y) = ( ,
?? ??
2
+ ?? ??
1
?? + ??
?? ??
2
+ ?? ??
1
?? + ??
)
Note : When line segment is divided
externally in the ratio m : n then ratio
becomes ( ,
?? ??
2
- ?? ??
1
?? - ??
?? ??
2
- ?? ??
1
?? - ??
)
Midpoint Formula : •
It is derived from the Section Formula.
The coordinates of a point which divides
the line segments in the ratio 1 : 1. Then,
(x, y) = ( , )
??
1
+ ??
2
2
??
1
+ ??
2
2
Centroid Formula : •
If (x
1
, y
1
) , (x
2
, y
2
) and (x
3
, y
3
) are the
vertices of a triangle then coordinates of
the centroid are :
( , )
??
1
+ ??
2
+ ??
3
3
??
1
+ ??
2
+ ??
3
3
Incentre Formula : •
If (x
1
, y
1
) , (x
2
, y
2
) and (x
3
, y
3
) are the
vertices of a triangle then coordinates of
the incenter are :
( , )
?? ??
1
+ ?? ??
2
+ ?? ??
3
?? + ?? + ??
?? ??
1
+ ?? ??
2
+ ?? ??
3
?? + ?? + ??
Slope of a line : •
If (x
1 ,
y
1
) , (x
2
, y
2
) are any two points on
line L, then the ratio is called the
??
2
- ??
1
??
2
- ??
1
slope of the line L.
Angle between two lines of slope m
1
•
and m
2,
respectively then
tan = ?
??
2
- ??
1
1 + ??
1
??
2
|
|
|
|
|
|
Intercept form of equation of a line : •
+ = 1,where ‘a’ is x intercept of line
??
??
??
??
and ‘b’ is y intercept of line .
Perpendicular distance from a point (x
1
, •
y
1
) to line ax + by + c = 0, then
D =
?? ??
1
+ ?? ??
1
+ ??
| |
??
2
+ ??
2
Distance between lines : • ?
a
1
x + b
1
y + c
1
= 0 and a
2
x + b
2
y + c
2
= 0 is
D =
??
1
- ??
2
??
2
+ ??
2
|
|
|
|
|
|
Equation of a circle •
+ + 2gx + 2fy + c = 0, where center ??
2
??
2
of the circle is (- g , - f) and radius =
??
2
+ ??
2
- ??
If point A (x
1
, y
1
) , B(x
2
, y
2
) and C (x
3
, y
3
) •
are the vertices of a triangle then area of
a triangle is :
1
2
??
1
( ??
2
- ??
3
) + ??
2
( ??
3
- ??
1
) + ??
3
( ??
1
- ??
2
)
| |
Equation of a circle - + - = ? ( ?? h )
2
( ?? ?? )
2
??
2
Where
( ) x, y coordinate of center point h , ?? ?
x x - coordinate of circle point ?
y y - coordinate of circle point ?
Variety Questions
Q.1. What is the area (in unit squares) of
the triangle enclosed by the graphs of
the equations 2x + 5y = 12, x + y = 3 and
the x-axis?
SSC CGL Tier II (03/02/2022)
(a) 2.5 (b) 3.5 (c) 3 (d) 4
Q.2. The equation of circle with centre (1
, – 2) and radius 4 cm is :
SSC CHSL 17/03/2020 (Afternoon)
(a) ??
2
+ ??
2
+ 2 ?? - 4 ?? = 16
(b) ??
2
+ ??
2
- 2 ?? + 4 ?? = 16
(c) ??
2
+ ??
2
+ 2 ?? - 4 ?? = 11
(d) ??
2
+ ??
2
- 2 ?? + 4 ?? = 11
Q.3. What is the area (in square units) of
the triangular region enclosed by the
graphs of the equations x + y = 3,
2x + 5y = 12 and the x axis?
SSC CGL Tier II (13/09/2019)
(a) 2 (b) 3 (c) 4 (d) 6
Practice Questions
SSC CHSL 2023 Tier - 1
Q.4. Find the radius of the circle
. ?? ² + ?? ² = 25
SSC CHSL 08/08/2023 (4th Shift)
(a) 25 units (b) 5 units
(c) 2 units (d) 12 units
SSC CGL 2022 Tier - 2
Q.5. Find the coordinates of the points
where the graph 57x 19y = 399 cuts -
the coordinate axis.
SSC CGL Tier II (07/03/2023)
(a) x-axis at (-7,0) and y-axis at (0,-21)
(b) x-axis at (-7,0) and y-axis at (0,21)
(c) x-axis at (7,0) and y-axis at (0,-21)
(d) x-axis at (7,0) and y-axis at (0,21)
SSC CGL 2021 Tier - 2
Q.6. The graph of the equation x = a
(a 0) is a ______. ?
SSC CGL Tier II (08/08/2022)
(a) line parallel to x axis
(b) line parallel to y axis
(c) line at an angle of 45 degree to y axis
(d) line at an angle of 45 degree to x axis
SSC CGL 2020 Tier - 2
Q.7. The graphs of the equations
4x + y = and x + y + = 0
1
3
8
3
1
2
3
4
5
2
and intersect at a point P . The point P
also lies on the graph of the equation:
SSC CGL Tier II (29/01/2022)
(a) 4x y 7 0 (b) x 3y 12 0 - + = - - =
(c) 3x y 7 0 (d) x 2y 0 - - = + - 5 =
Q.8. What is the area (in unit squares) of
the region enclosed by the graphs of the
equations 2x – 3y + 6 = 0, 4x + y = 16 and
y = 0 ?
SSC CGL Tier II (29/01/2022)
(a) 11.5 (b) 14 (c) 10.5 (d) 12
Q.9. The graphs of the equations
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Pinnacle Day: 89th Coordinate Geometry
7x + 11y = 3 and 8x + y = 15 intersect at
the point P , which also lies on the graph
of the equation.
SSC CGL Tier II (03/02/2022)
(a) 2x + y = 2 (b) 2x – y = 1
(c) 3x + 5y = 1 (d) 3x + 2y = 3
SSC CGL 2019 Tier - 2
Q.10. The graph of the equations
3x – 20y – 2 = 0 and 11x – 5y + 61 = 0
intersect at P(a, b). What is the value of
? ( ??
2
+ ??
2
- ???? ) / ( ??
2
- ??
2
+ ???? )
SSC CGL Tier II (15/11/2020)
(a) (b) (c) (d)
37
35
31
41
5
7
41
31
Q.11. The area (in sq units) of the
triangle formed by the graphs of
equations 8x + 3y = 24 , 2x + 8 = y and
the x-axis is:
SSC CGL Tier II (15/11/2020)
(a) 28 (b) 14 (c) 15 (d)24
Q.12. The graph of the linear equation
3x – 2y = 8 and 4x + 3y = 5 intersect at
the point ( ). What is the value of a, ß
(2 ? a - ß)
SSC CGL Tier II (16/11/2020)
(a) 4 (b) 6 (c) 3 (d) 5
Q.13. What is the re?ection of the point
(5, – 3) in the line Y = 3 ?
SSC CGL Tier II (18/11/2020)
(a) (5, -6) (b) ( -5, 3) (c) (5, 9) (d) (5, 3)
Q.14. The graphs of the linear equations
4x – 2y = 10 and 4x + ky = 2 intersect at a
point (a, 4). The value of k is equal to:
SSC CGL Tier II ( 18/11/2020)
(a) 3 (b) – 3 (c) – 4 (d) 4
SSC CGL 2018 Tier - 2
Q.15. The graphs of the equations
3x + y – 5 = 0 and 2x – y – 5 = 0 intersect
at the point P( ). What is the value of a, ß
( ) ? 3 a + ß
SSC CGL Tier II (11/09/2019)
(a) 4 (b) - 4 (c)3 (d) 5
Q.16. The graph of the equation
x – 7y = – 42, intersects the y -axis at
P( ) and the graph of 6x + y – 15 = 0, a , ß
intersects the x-axis at Q( ). What is ? , d
the value of a + ß + ? + d ?
SSC CGL Tier II (11/09/2019)
(a) (b) 6 (c) (d) 5
17
2
9
2
Q.17. The point of intersection of the
graphs of the equations 3x – 5y = 19 and
3y – 7x + 1 = 0 is P( ) . What is the a , ß
value of ( ) ? 3 a - ß
SSC CGL Tier II (12/09/2019)
(a) - 2 (b) –1 (c) 1 (d) 0
Q.18. The graphs of the equations
2x + 3y = 11 and x – 2y + 12 = 0
intersects at P(
1 , 1
) and the graph of ?? ??
the equation x – 2y + 12 = 0 intersects
the x-axis at Q( ). What is the value ??
2
, ??
2
of ( ) ? ??
1
- ??
2
+ ??
1
+ ??
2
SSC CGL Tier II (12/09/2019)
(a) 13 (b) - 11 (c) 15 (d) -9
Q.19. The graph of the equations
5x – 2y + 1 = 0 and 4y – 3x + 5 = 0,
intersect at the point P( ). What is the a, ß
value of ( ) ? 2 a - 3 ß
SSC CGL Tier II (13/09/2019)
(a) 4 (b) 6 (c) - 4 (d) - 3
Answer Key :-
1.(c) 2.(d) 3.(b) 4.(b)
5.(c) 6.(b) 7.(c) 8.(b)
9.(c) 10.(b) 11.(a) 12.(d)
13.(c) 14.(c) 15.(d) 16.(a)
17.(b) 18.(c) 19.(a)
Solutions :-
Sol.1.(c) Intersection point of 2x + 5y
= 12 and x + y = 3 is (1 , 2)
Intersection point of 2x + 5y = 12 and
y = 0 is (6 , 0)
Intersection point of x + y = 3 and y = 0
is (3 , 0)
Area of triangle = ? +
1
2
[ ??
1
( ??
2
- ??
3
)
+ ] ? ??
2
( ??
3
- ??
1
) ??
3
( ??
1
- ??
2
)
? ? [1(0 – 0) + 6(0 – 2) + 3(2 – 0)] ?
1
2
? = 3
1
2
[- 12 + 6 ] | | ????????
2
Sol.2.(d)
Center = (1, –2) and radius = 4 cm
Equation of circle
(x – 1)
2
+ (y + 2)
2
= (4)
2
?
x
2
+ 1 – 2x + y
2
+ 4 + 4y = 16
x
2
– 2x + y
2
+ 4y = 16 – 5 ?
x
2
+ y
2
– 2x + 4y = 11 ?
Sol.3.(b) In x + y = 3
put x = 0 ? ?? = 3
put y = 0 ? ?? = 3
Line formed by the equation will be AB
In 2x + 5y = 12
put x = 0 ? ?? = 2 . 4
put y = 0 ? ?? = 6
Line formed by the equation will be CD
AB and CD will intersect at point E.
Given, x + y = 3 ………(1)
and 2x + 5y = 12 …….(2)
Multiply equation (1) by 2 and subtract it
from equation (2)
? 3 ?? = 6 ? ?? = 2
Put the value of y in any of the equations
x + 2 = 3 ? ?? = 1
So, E(x , y) = (1 , 2)
Area formed by the the equations
x + y = 3 , 2x + 5y = 12 and the x axis is
the shaded region or ? ??????
Now, AC = OC – OA = 6 – 3 = 3
Height of the triangle will be the y
coordinate of point E = 2
Required area = base Height
1
2
× ×
= 3 2 = 3 square units.
1
2
× ×
Sol.4.(b) Equation of circle passing
through origin :-
+ = …..( = radius) ??
2
??
2
??
2
??
Here , + = radius = 5 units ??
2
??
2
25 ?
Sol.5.(c) 57x 19y = 399 -
At x-axis y = 0 ?
57x 19y = 399 x = = 7 - ?
399
57
At y-axis x = 0 ?
57x 19y = 399 y = = 21 - ?
- 399
19
-
So , the graph 57x 19y = 399 cuts the -
coordinate axis ,
x-axis at (7,0) and y-axis at (0, 21) -
Sol.6.(b)
The graph of the equation x = a is a line
parallel to the y axis.
Sol.7.(c) 4x + y =
1
3
8
3
12x + y = 8 ………..e.q .(1)
x + y + = 0
1
2
3
4
5
2
2x + 3y = 10 ………..e.q .(2) -
Multiplying (1) by 3 and subtracting from
(2) we get,
– 34x = – 34 , x = 1, ?
Putting x = 1 in e.q .(1) , we get
y = – 4
Checking these values of x and y we ?nd
only option (c) satis?es these values.
Sol.8.(b) Intersection point of 2x – 3y +
6 = 0 and 4x + y = 16 is (3 , 4)
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Page 3
Pinnacle Day: 89th Coordinate Geometry
Coordinate Geometry
Coordinate geometry is a system of
geometry where the position of points on
the plane is described by using an
ordered pair of numbers. The x-
coordinate and the y- coordinate of a
point taken together is known as
coordinates of the point
Key points:
Coordinate Axes : The perpendicular •
lines in the cartesian system are x axis
and y axis . Both of them together are
known as coordinate axes.
Origin : The point of intersection of •
coordinate axes is origin. The
coordinates of origin are (0 , 0).
Abscissa : The x - coordinate is called •
abscissa. It is the perpendicular distance
of a point from y axis.
Ordinate : The y-coordinate is called •
ordinate . It is the perpendicular distance
of a point from x-axis.
If y = a, where a is constant then y = a •
denotes a straight line parallel to x-axis.
If x = a, where a is constant then x = a •
denotes a straight line parallel to y-axis.
For passing a line through the origin, •
one of the coordinates of the given
equation must be (0, 0).
Important Formulae :
Distance Formula : If points P(x
1
, y
1
), •
Q(x
2 ,
y
2
) lie on the plane xy then,
PQ = ( ??
2
- ??
1
)
2
+ ( ??
2
- ??
1
)
2
Section Formula : •
The coordinates of a point which divides
the line segment joined by two distinct
points (x
1
, y
1
) and (x
2
, y
2
) in the ratio
m : n are :
(x , y) = ( ,
?? ??
2
+ ?? ??
1
?? + ??
?? ??
2
+ ?? ??
1
?? + ??
)
Note : When line segment is divided
externally in the ratio m : n then ratio
becomes ( ,
?? ??
2
- ?? ??
1
?? - ??
?? ??
2
- ?? ??
1
?? - ??
)
Midpoint Formula : •
It is derived from the Section Formula.
The coordinates of a point which divides
the line segments in the ratio 1 : 1. Then,
(x, y) = ( , )
??
1
+ ??
2
2
??
1
+ ??
2
2
Centroid Formula : •
If (x
1
, y
1
) , (x
2
, y
2
) and (x
3
, y
3
) are the
vertices of a triangle then coordinates of
the centroid are :
( , )
??
1
+ ??
2
+ ??
3
3
??
1
+ ??
2
+ ??
3
3
Incentre Formula : •
If (x
1
, y
1
) , (x
2
, y
2
) and (x
3
, y
3
) are the
vertices of a triangle then coordinates of
the incenter are :
( , )
?? ??
1
+ ?? ??
2
+ ?? ??
3
?? + ?? + ??
?? ??
1
+ ?? ??
2
+ ?? ??
3
?? + ?? + ??
Slope of a line : •
If (x
1 ,
y
1
) , (x
2
, y
2
) are any two points on
line L, then the ratio is called the
??
2
- ??
1
??
2
- ??
1
slope of the line L.
Angle between two lines of slope m
1
•
and m
2,
respectively then
tan = ?
??
2
- ??
1
1 + ??
1
??
2
|
|
|
|
|
|
Intercept form of equation of a line : •
+ = 1,where ‘a’ is x intercept of line
??
??
??
??
and ‘b’ is y intercept of line .
Perpendicular distance from a point (x
1
, •
y
1
) to line ax + by + c = 0, then
D =
?? ??
1
+ ?? ??
1
+ ??
| |
??
2
+ ??
2
Distance between lines : • ?
a
1
x + b
1
y + c
1
= 0 and a
2
x + b
2
y + c
2
= 0 is
D =
??
1
- ??
2
??
2
+ ??
2
|
|
|
|
|
|
Equation of a circle •
+ + 2gx + 2fy + c = 0, where center ??
2
??
2
of the circle is (- g , - f) and radius =
??
2
+ ??
2
- ??
If point A (x
1
, y
1
) , B(x
2
, y
2
) and C (x
3
, y
3
) •
are the vertices of a triangle then area of
a triangle is :
1
2
??
1
( ??
2
- ??
3
) + ??
2
( ??
3
- ??
1
) + ??
3
( ??
1
- ??
2
)
| |
Equation of a circle - + - = ? ( ?? h )
2
( ?? ?? )
2
??
2
Where
( ) x, y coordinate of center point h , ?? ?
x x - coordinate of circle point ?
y y - coordinate of circle point ?
Variety Questions
Q.1. What is the area (in unit squares) of
the triangle enclosed by the graphs of
the equations 2x + 5y = 12, x + y = 3 and
the x-axis?
SSC CGL Tier II (03/02/2022)
(a) 2.5 (b) 3.5 (c) 3 (d) 4
Q.2. The equation of circle with centre (1
, – 2) and radius 4 cm is :
SSC CHSL 17/03/2020 (Afternoon)
(a) ??
2
+ ??
2
+ 2 ?? - 4 ?? = 16
(b) ??
2
+ ??
2
- 2 ?? + 4 ?? = 16
(c) ??
2
+ ??
2
+ 2 ?? - 4 ?? = 11
(d) ??
2
+ ??
2
- 2 ?? + 4 ?? = 11
Q.3. What is the area (in square units) of
the triangular region enclosed by the
graphs of the equations x + y = 3,
2x + 5y = 12 and the x axis?
SSC CGL Tier II (13/09/2019)
(a) 2 (b) 3 (c) 4 (d) 6
Practice Questions
SSC CHSL 2023 Tier - 1
Q.4. Find the radius of the circle
. ?? ² + ?? ² = 25
SSC CHSL 08/08/2023 (4th Shift)
(a) 25 units (b) 5 units
(c) 2 units (d) 12 units
SSC CGL 2022 Tier - 2
Q.5. Find the coordinates of the points
where the graph 57x 19y = 399 cuts -
the coordinate axis.
SSC CGL Tier II (07/03/2023)
(a) x-axis at (-7,0) and y-axis at (0,-21)
(b) x-axis at (-7,0) and y-axis at (0,21)
(c) x-axis at (7,0) and y-axis at (0,-21)
(d) x-axis at (7,0) and y-axis at (0,21)
SSC CGL 2021 Tier - 2
Q.6. The graph of the equation x = a
(a 0) is a ______. ?
SSC CGL Tier II (08/08/2022)
(a) line parallel to x axis
(b) line parallel to y axis
(c) line at an angle of 45 degree to y axis
(d) line at an angle of 45 degree to x axis
SSC CGL 2020 Tier - 2
Q.7. The graphs of the equations
4x + y = and x + y + = 0
1
3
8
3
1
2
3
4
5
2
and intersect at a point P . The point P
also lies on the graph of the equation:
SSC CGL Tier II (29/01/2022)
(a) 4x y 7 0 (b) x 3y 12 0 - + = - - =
(c) 3x y 7 0 (d) x 2y 0 - - = + - 5 =
Q.8. What is the area (in unit squares) of
the region enclosed by the graphs of the
equations 2x – 3y + 6 = 0, 4x + y = 16 and
y = 0 ?
SSC CGL Tier II (29/01/2022)
(a) 11.5 (b) 14 (c) 10.5 (d) 12
Q.9. The graphs of the equations
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Pinnacle Day: 89th Coordinate Geometry
7x + 11y = 3 and 8x + y = 15 intersect at
the point P , which also lies on the graph
of the equation.
SSC CGL Tier II (03/02/2022)
(a) 2x + y = 2 (b) 2x – y = 1
(c) 3x + 5y = 1 (d) 3x + 2y = 3
SSC CGL 2019 Tier - 2
Q.10. The graph of the equations
3x – 20y – 2 = 0 and 11x – 5y + 61 = 0
intersect at P(a, b). What is the value of
? ( ??
2
+ ??
2
- ???? ) / ( ??
2
- ??
2
+ ???? )
SSC CGL Tier II (15/11/2020)
(a) (b) (c) (d)
37
35
31
41
5
7
41
31
Q.11. The area (in sq units) of the
triangle formed by the graphs of
equations 8x + 3y = 24 , 2x + 8 = y and
the x-axis is:
SSC CGL Tier II (15/11/2020)
(a) 28 (b) 14 (c) 15 (d)24
Q.12. The graph of the linear equation
3x – 2y = 8 and 4x + 3y = 5 intersect at
the point ( ). What is the value of a, ß
(2 ? a - ß)
SSC CGL Tier II (16/11/2020)
(a) 4 (b) 6 (c) 3 (d) 5
Q.13. What is the re?ection of the point
(5, – 3) in the line Y = 3 ?
SSC CGL Tier II (18/11/2020)
(a) (5, -6) (b) ( -5, 3) (c) (5, 9) (d) (5, 3)
Q.14. The graphs of the linear equations
4x – 2y = 10 and 4x + ky = 2 intersect at a
point (a, 4). The value of k is equal to:
SSC CGL Tier II ( 18/11/2020)
(a) 3 (b) – 3 (c) – 4 (d) 4
SSC CGL 2018 Tier - 2
Q.15. The graphs of the equations
3x + y – 5 = 0 and 2x – y – 5 = 0 intersect
at the point P( ). What is the value of a, ß
( ) ? 3 a + ß
SSC CGL Tier II (11/09/2019)
(a) 4 (b) - 4 (c)3 (d) 5
Q.16. The graph of the equation
x – 7y = – 42, intersects the y -axis at
P( ) and the graph of 6x + y – 15 = 0, a , ß
intersects the x-axis at Q( ). What is ? , d
the value of a + ß + ? + d ?
SSC CGL Tier II (11/09/2019)
(a) (b) 6 (c) (d) 5
17
2
9
2
Q.17. The point of intersection of the
graphs of the equations 3x – 5y = 19 and
3y – 7x + 1 = 0 is P( ) . What is the a , ß
value of ( ) ? 3 a - ß
SSC CGL Tier II (12/09/2019)
(a) - 2 (b) –1 (c) 1 (d) 0
Q.18. The graphs of the equations
2x + 3y = 11 and x – 2y + 12 = 0
intersects at P(
1 , 1
) and the graph of ?? ??
the equation x – 2y + 12 = 0 intersects
the x-axis at Q( ). What is the value ??
2
, ??
2
of ( ) ? ??
1
- ??
2
+ ??
1
+ ??
2
SSC CGL Tier II (12/09/2019)
(a) 13 (b) - 11 (c) 15 (d) -9
Q.19. The graph of the equations
5x – 2y + 1 = 0 and 4y – 3x + 5 = 0,
intersect at the point P( ). What is the a, ß
value of ( ) ? 2 a - 3 ß
SSC CGL Tier II (13/09/2019)
(a) 4 (b) 6 (c) - 4 (d) - 3
Answer Key :-
1.(c) 2.(d) 3.(b) 4.(b)
5.(c) 6.(b) 7.(c) 8.(b)
9.(c) 10.(b) 11.(a) 12.(d)
13.(c) 14.(c) 15.(d) 16.(a)
17.(b) 18.(c) 19.(a)
Solutions :-
Sol.1.(c) Intersection point of 2x + 5y
= 12 and x + y = 3 is (1 , 2)
Intersection point of 2x + 5y = 12 and
y = 0 is (6 , 0)
Intersection point of x + y = 3 and y = 0
is (3 , 0)
Area of triangle = ? +
1
2
[ ??
1
( ??
2
- ??
3
)
+ ] ? ??
2
( ??
3
- ??
1
) ??
3
( ??
1
- ??
2
)
? ? [1(0 – 0) + 6(0 – 2) + 3(2 – 0)] ?
1
2
? = 3
1
2
[- 12 + 6 ] | | ????????
2
Sol.2.(d)
Center = (1, –2) and radius = 4 cm
Equation of circle
(x – 1)
2
+ (y + 2)
2
= (4)
2
?
x
2
+ 1 – 2x + y
2
+ 4 + 4y = 16
x
2
– 2x + y
2
+ 4y = 16 – 5 ?
x
2
+ y
2
– 2x + 4y = 11 ?
Sol.3.(b) In x + y = 3
put x = 0 ? ?? = 3
put y = 0 ? ?? = 3
Line formed by the equation will be AB
In 2x + 5y = 12
put x = 0 ? ?? = 2 . 4
put y = 0 ? ?? = 6
Line formed by the equation will be CD
AB and CD will intersect at point E.
Given, x + y = 3 ………(1)
and 2x + 5y = 12 …….(2)
Multiply equation (1) by 2 and subtract it
from equation (2)
? 3 ?? = 6 ? ?? = 2
Put the value of y in any of the equations
x + 2 = 3 ? ?? = 1
So, E(x , y) = (1 , 2)
Area formed by the the equations
x + y = 3 , 2x + 5y = 12 and the x axis is
the shaded region or ? ??????
Now, AC = OC – OA = 6 – 3 = 3
Height of the triangle will be the y
coordinate of point E = 2
Required area = base Height
1
2
× ×
= 3 2 = 3 square units.
1
2
× ×
Sol.4.(b) Equation of circle passing
through origin :-
+ = …..( = radius) ??
2
??
2
??
2
??
Here , + = radius = 5 units ??
2
??
2
25 ?
Sol.5.(c) 57x 19y = 399 -
At x-axis y = 0 ?
57x 19y = 399 x = = 7 - ?
399
57
At y-axis x = 0 ?
57x 19y = 399 y = = 21 - ?
- 399
19
-
So , the graph 57x 19y = 399 cuts the -
coordinate axis ,
x-axis at (7,0) and y-axis at (0, 21) -
Sol.6.(b)
The graph of the equation x = a is a line
parallel to the y axis.
Sol.7.(c) 4x + y =
1
3
8
3
12x + y = 8 ………..e.q .(1)
x + y + = 0
1
2
3
4
5
2
2x + 3y = 10 ………..e.q .(2) -
Multiplying (1) by 3 and subtracting from
(2) we get,
– 34x = – 34 , x = 1, ?
Putting x = 1 in e.q .(1) , we get
y = – 4
Checking these values of x and y we ?nd
only option (c) satis?es these values.
Sol.8.(b) Intersection point of 2x – 3y +
6 = 0 and 4x + y = 16 is (3 , 4)
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Pinnacle Day: 89th Coordinate Geometry
Intersection point of 2x – 3y + 6 = 0 and
y = 0 is ( – 3 , 0)
Intersection point of 4x + y = 16 and
y = 0 is (4 , 0)
Area of triangle = ? +
1
2
[ ??
1
( ??
2
- ??
3
)
+ ] ? ??
2
( ??
3
- ??
1
) ??
3
( ??
1
- ??
2
)
? ? [3(0 – 0) – 3(0 – 4) + 4(4 – 0)] ?
1
2
? ? [ ] ? = 14
1
2
12 + 16 ????????
2
Short Trick :-
Required area = × 7 units × 4 units =
1
2
14 ????????
2
Sol.9.(c) 7x + 11y = 3 …..(1)
8x + y = 15 ……(2)
On solving eqn. (1) and (2) , we get
y = – 1 and x = 2
If we put the obtained value of x and y in
options given, then only option (c) gets
satis?ed.
Sol.10.(b) The point where both the lines
meet is the only point which satis?es
both the equations.
On solving both the e.q . , we get
a = 6 and b = 1
2 2
( ??
2
+ ??
2
- ???? ) / ( ?? - ?? + ???? )
= =
36 + 1 - 6
36 - 1 + 6
31
41
Sol.11.(a) Points of intersection
By 8x + 3y = 24 and 2x + 8 = y is (0 , 8)
y = 0 and 2x + 8 = y is ( 4 ,0) -
y = 0 and 8x + 3y = 24 is (3 ,0)
Base = 7 and height = 8
Area = 8 7 = 28
1
2
× × ????????
2
Sol.12.(d) 3x – 2y = 8 and 4x + 3y = 5
On solving both the e.q . , we get
= 2 , = – 1 (2 = 5 a ß ? a - ß)
Sol.13.(c) y = 3 is parallel to x axis so on
re?ection x axis will not change
Perpendicular distance of (5 , – 3) from
y = 3 = 6 units
6 units on the other side of the line
(y = 3 ) = 3 + 6 = 9
So the point = (5 , 9)
Sol.14.(c) They intersect at y = 4
So , 4x – 2(4) = 10 x = 4.5 ?
4x + ky = 2 4(4.5) + k(4) = 2 ?
k = – 4
Sol.15.(d) 3x + y = 5 ………e.q .(1)
2x – y = 5 ……..e.q .(2)
Adding e.q . (1) and (2) , we get
x = 2 ?
Put the value of x in e.q . (1) , we get
3x + y = 5 3(2) + y = 5 y = 1 ? ? -
So, intersecting point of the given lines
{P( )} = (2 , – 1) a , ß
= 5 ? ( 3 a + ß) = { 3 ( 2 ) + (- 1 )}
Sol.16.(a) Given ,
x – 7y = – 42 ………e.q .(1)
6x + y – 15 = 0 ………e.q .(2)
On y axis, x = 0
Put the value of x in e.q . (1) , we get
0 – 7y = – 42 ? ? ?? = 6
? ?? (a , ß) = ( 0 , 6 )
On x axis , y = 0
Put the value of y in e.q . (2) , we get
6x + 0 = 15 x = ? ?
5
2
? ?? (? , d) = (
5
2
, 0 )
? ??h?? ?????????? ???? a + ß + ? + d
= 0 + 6 + =
5
2
+ 0
17
2
Sol.17.(b) Given,
3x – 5y =19 ……e.q .(1)
and 3y – 7x + 1= 0
7x – 3y = 1 ……..e.q .(2)
Multiply e.q :- (1) by 3 and e.q . (2) by 5 and
subtract e.q :- (2) from e.q .(1) , we get
? 26 ?? = - 52 ? ?? = - 2
Putting the value of x in e.q . (1) ,
3( – 2) – 5y = 19
? ?? = - 5 ? ?? (a , ß) = (- 2 , - 5 )
( ) 3 a - ß ? 3 (- 2 ) - (- 5 ) = - 1
Sol.18.(c) 2x + 3y =11 ……….e.q .(1)
and x – 2y + 12 = 0
2y – x = 12 ………..e.q . (2)
Multiply e.q .(2) by 2 and then add both
the equation ,
and x = – 2 ? 7 ?? = 35 ? ?? = 5
1
,
1
? ?? ( ?? ?? ) = (- 2 , 5 )
Now,
x – 2y + 12 = 0
At x axis y = 0 ? ?? = - 12
Q(
2 , 2
) = ( – 12 , 0) ?? ??
( ) ??
1
- ??
2
+ ??
1
+ ??
2
{ – 2 – ( – 12) + 5 + 0 } = 15 ?
Sol.19.(a)
5x – 2y + 1 = 0 or 2y – 5x = 1…e.q .(1)
4y – 3x + 5 = 0 or 3x – 4y = 5 …e.q .(2)
Multiply eq(1) by 2 and add it in eq (2)
- 7 ?? = 7 ? ?? = - 1
Put this value in any of the equation
5( – 1) – 2y + 1 = 0
? ?? = - 2
? ?? (a , ß) = (- 1 , - 2 )
= 4 ? ( 2 a - 3 ß) = { 2 (- 1 ) - 3 (- 2 )
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FAQs on SSC CGL Previous Year Questions (2023 - 18): Coordinate Geometry - SSC CGL Previous Year Papers
| 1. What are the key concepts of Coordinate Geometry that are important for SSC CGL exams? |  |
Ans. Key concepts include the Cartesian coordinate system, distance formula, section formula, area of triangles, slopes of lines, equations of lines (slope-intercept and point-slope forms), and properties of various geometric shapes like circles, triangles, and rectangles in the coordinate plane.
| 2. How do you calculate the distance between two points in Coordinate Geometry? | |
Ans. The distance \(d\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) can be calculated using the distance formula:
\[ d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} \]
| 3. What is the section formula in Coordinate Geometry and how is it applied? | |
Ans. The section formula is used to find the coordinates of a point that divides a line segment joining two points \((x_1, y_1)\) and \((x_2, y_2)\) in the ratio \(m:n\). The coordinates of the dividing point \(P\) are given by:
\[ P\left(\frac{mx_2 + nx_1}{m+n}, \frac{my_2 + ny_1}{m+n}\right) \]
| 4. How do you find the area of a triangle given its vertices in Coordinate Geometry? | |
Ans. The area \(A\) of a triangle with vertices at \((x_1, y_1)\), \((x_2, y_2)\), and \((x_3, y_3)\) can be found using the formula:
\[ A = \frac{1}{2} \left| x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2) \right| \]
| 5. What are the various forms of the equation of a line in Coordinate Geometry, and when are they used? | |
Ans. The main forms of the equation of a line include:
1. Slope-intercept form: \(y = mx + c\) (used when the slope and y-intercept are known).
2. Point-slope form: \(y - y_1 = m(x - x_1)\) (used when a point on the line and the slope are known).
3. Two-point form: \(y - y_1 = \frac{y_2 - y_1}{x_2 - x_1}(x - x_1)\) (used when two points on the line are known).
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SSC CGL Previous Year Questions (2023 - 18): Coordinate Geometry SSC CGL
The "SSC CGL Previous Year Questions (2023 - 18): Coordinate Geometry SSC CGL Questions" guide is a valuable resource for all aspiring students preparing for the
SSC CGL exam. It focuses on providing a wide range of practice questions to help students gauge
their understanding of the exam topics. These questions cover the entire syllabus, ensuring comprehensive preparation.
The guide includes previous years' question papers for students to familiarize themselves with the exam's format and difficulty level.
Additionally, it offers subject-specific question banks, allowing students to focus on weak areas and improve their performance.
Study SSC CGL Previous Year Questions (2023 - 18): Coordinate Geometry on the App
Students of SSC CGL can study SSC CGL Previous Year Questions (2023 - 18): Coordinate Geometry alongwith tests & analysis from the EduRev app,
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The content of SSC CGL Previous Year Questions (2023 - 18): Coordinate Geometry is prepared as per the latest SSC CGL syllabus.
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