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Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making PDF Download

The X – Y Plane

The number lines, when drawn as shown in X – Y plane below, are called "axes". The horizontal number line
is called the "x-axis", the vertical one is the y-axis.

Abscissa is the x–coordinate of a point can be defined either as its distance along the x–axis, or as its
perpendicular distance from the y–axis.

Ordinate is the y-coordinate of a point can be defined either as its distance along the y-axis, or as its
perpendicular distance from the x-axis.

In the figure given below, OX and OY are two straight lines which are perpendicular to each other and which
intersect at the point O. OX is known as the x–axis, and OY is known as the y–axis. You can see that the two
axes divide the plane into four regions as above. The four regions are known as Quadrants and are named I
Quadrant, II Quadrant, III Quadrant and IV Quadrant as shown.

Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
(A) Co-ordinate of the origin is (0, 0).
(B) Any point on the x axis can be taken as (a, 0)
(C) Any point on the y axis can be taken as (0, b)


Ex.1 In which quadrant is (x, y), such that x y < 0?

Sol. The points (x, y), with xy < 0 means the product of abscissa and ordinate should be negative which can
be possible only when one is positive and other is negative, such as (– 2, 5) or (4, – 6) and this will lie
in the Quadrants II and IV.

(i) Distance formula:
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

Ex.2 A (a, 0) and B (3a, 0) are the vertices of an equilateral triangle ABC. What are the coordinates of
 C?
 (1) (a, a√3) 

(2) (a√3, 2a) 

(3) (a√3, 0) 

(4) (2a, + a√3) 

(5) None of these
 

Sol. Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
Now, the vertex C will be such that AC = BC = 2a
∴ if (x, y) are the co-ordinates of C
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
or x = 2a.
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making


(ii) Section formula:
The point which divides the join of two distinct points A (x1, y1)
and B (x2, y2) in the ratio m1 : m2 Internally, has the co-ordinates
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
In particular, the mid-point of the segment joining A (x1, y1) and B (x2, y2) has the co-ordinates
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making


Ex.3 Find the points A and B which divide the join of points (1, 3) and (2, 7) in ratio 3 : 4 both
 internally & externally respectively .

Sol.

Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making      Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making



(iii) Centroid and Incentre formulae:
 

Centroid: It is the point of intersection of the medians of a triangle.
 

Incentre: It is the point of intersection of the internal angle bisectors of the angles of a triangle.

If A Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making be the vertices of a triangle, then its centroid is given by
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making  and the incentre by,  Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making


Where a = | BC |, b = | CA | and c = | AB |.


Ex.4 If (2, 3), (3, a), (b, – 2) are the vertices of the triangle whose centroid is (0, 0), then find the
 value of a and b respectively.
 (1) – 1, – 4 

(2) – 2, – 5 

(3) – 1, – 6 

(4) – 1, – 5 

(5) None of these

 

Sol. x co-ordinate of the centroid = Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
y co-ordinate of the centroid = Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
a = – 1             Answer: (4)


(iv) Area of triangle:
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision MakingCoordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

 

Ex.5 A triangle has vertices A (2, 2), B (5, 2) and C (5, 6). What type of triangle it is ?


Sol. By the distance formula
Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making
According to Pythagorean if the sum of the square of two sides are equal to the square of the third side
then the triangle is a right-angled triangle.
d(AC)2 = d(AB)2 + d(BC)2
(5)2 = (3)2 + (4)2
Therefore, the triangle is right-angled.


Important


Coordinate Geometry - Introduction and Examples (with Solutions) | UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

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FAQs on Coordinate Geometry - Introduction and Examples (with Solutions) - UPSC Prelims Paper 2 CSAT - Quant, Verbal & Decision Making

1. What is coordinate geometry and how is it used in mathematics?
Ans. Coordinate geometry is a branch of mathematics that combines algebra and geometry. It involves plotting points on a coordinate plane and using mathematical equations to determine the relationships between these points. It is widely used in various fields such as engineering, physics, and computer science to solve problems involving shapes, distances, and equations of lines and curves.
2. How do you find the distance between two points in coordinate geometry?
Ans. To find the distance between two points in coordinate geometry, you can use the distance formula. The distance formula states that the distance between two points (x1, y1) and (x2, y2) is given by the square root of [(x2 - x1)^2 + (y2 - y1)^2]. By substituting the coordinates of the two points into this formula, you can calculate the distance between them.
3. What is the equation of a straight line in coordinate geometry?
Ans. The equation of a straight line in coordinate geometry is usually represented in the form y = mx + c, where m is the slope of the line and c is the y-intercept. The slope indicates the steepness of the line, while the y-intercept represents the point where the line intersects the y-axis. By knowing the slope and y-intercept, you can easily plot the line on a coordinate plane or determine its equation.
4. How can coordinate geometry be used to solve real-life problems?
Ans. Coordinate geometry can be used to solve various real-life problems. For example, it can be used to determine the shortest distance between two locations on a map, calculate the area of irregular shapes, find the equation of a line that best fits a set of data points, or analyze the motion of objects in physics. By applying the principles of coordinate geometry, you can solve practical problems efficiently and accurately.
5. Can coordinate geometry be used to find the intersection point of two lines?
Ans. Yes, coordinate geometry can be used to find the intersection point of two lines. To find the intersection point, you need to equate the equations of the two lines and solve the resulting system of equations. The solution to this system of equations will give you the coordinates of the point where the two lines intersect. This method is commonly used in mathematics and physics to find the intersection of lines or determine the common solutions to linear equations.
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