Table of contents  
Introduction  
Coordinate Geometry Formulas  
Equations of a Line  
Knowledge of Quadrants  
Solved Examples  
CAT Coordinate Geometry Questions 
Coordinate Geometry holds significant importance in the CAT Quantitative Ability (QA) section. Typically, problems in Coordinate Geometry are not overly challenging, making it crucial not to overlook them in the CAT exam. Practicing a sufficient number of CAT Coordinate Geometry questions enhances your ability to tackle such problems effortlessly during the actual exam. In this article, we will delve into essential Coordinate Geometry questions for CAT QA, offering valuable practice.
(1) Distance Formula:
To Calculate Distance Between Two Points:
Let the two points be A and B, having coordinates to be (x_1,y_1) and (x_2,y_2) respectively.
Thus, the distance between two points is
distance =
(2) Midpoint Theorem:
To Find Midpoint of a Line Connecting Two Points:
Consider the same points A and B, having coordinates to be (x1,y1) and (x2,y2) respectively. Let M(x,y) be the midpoint of lying on the line connecting these two points A and B. The coordinates of this point are –
(3) Angle Formula:
To Find The Angle Between Two Lines:
Consider two straight lines and , with given slopes as m_1 and m_2 respectively.
Let “θ” be the angle between these two lines, we can then represent the angle between them as
(4) Section Formula:
To Find a Point which divides a line into m:n Ratio using:
Consider a two straight lines having coordinates (x1,y1) & (x2,y2) respectively. Let a point which divides the line in some ratio as m:n, then the coordinates of this point are
When the ratio m:n is internal:
When the ratio m:n is external:
(5) Area of a Triangle in Cartesian Plane:
We can compute the area of a triangle in Cartesian Geometry if we know all the coordinates of all three vertices. If coordinats are (x1,y1),(x2,y2) and (x3,y3) then area will be:
Now, Let us have a look at some more formulas for coordinate geometry. We will use the below picture as a reference for the formulas.
Example 1: What is the distance between the points A (3,8) and B(2,7) ?
a) 5√2
b) 5
c) 5√10
d) 10√2
 The distance between 2 points (x1, y1) and (x2, y2) is given as
 Sqrt ((x2x1)^{2} + (y2y1)^{2})
 Hence, required distance = sqrt((23)^{2} + (78)^{2}) = 5√10
Example 2: The points of intersection of three lines 2X + 3Y – 5=0 and 5X – 7Y + 2=0 and 9X – 5Y – 4 = 0
a) Form a triangle
b) Are on lines perpendicular to each other
c) Are on lines parallel to each other
d) Are coincident
To solve the question above, we should remember the properties of the lines for being parallel, perpendicular or intersecting:
 Two lines are parallel to each other if their slopes are equal
 Two lines are perpendicular if the product of slopes is 1
 Lines are coincident if they at least have one point which satisfies all the equation.
 The three lines can be expressed in the y=mx + c format as:
Y = (5/3) – (2X/3), Y = (5X/7) + (2/7) , Y = (9X/5) – (4/5) Therefore, the slopes of the three lines are 2/3, 5/7, 9/5 and their Y intercepts are 5/3, 2/7 and 4/5 respectively.
 We see above that the product of slopes of none of the lines is 1. Thus, lines are not perpendicular to each other.
 Also, slopes of the no two lines is same. Thus, lines are not parallel to each other.
 Solving the first two equations we get X=1 and Y = 1. If we substitute (1,1) in the third equation Y=(9X/5 – 4/5), we find that it also satisfies the equation. This suggests that the three lines intersect at a common point and hence coincident.
Example 3: The area of the triangle whose vertices are (a + 1, a + 1), (a, a) and (a+2, a) is
a) a^{3}
b) 1
c) 2a
d) 2^{1/2}
 Let a = 0, Thus the three vertices of the triangle becomes (1, 1) (0, 0) and (2, 0)
 If we look at the below figure, Area = ½ * base * height = ½ * 2 * 1 = 1
 Imp: The main point to note here is that area will be independent of a.
Example 4: Consider a triangle drawn on the X – Y plane with its three vertices of (41,0) , (0,41) and (0,0), each vertex being represented by its (X,Y) coordinates. The number of points with integer coordinates inside the triangle (excluding all the points on the boundary) is:
a) 12
b) 8
c) 6
d) 4
 Equation of the line will be of the form => x + y = 41.
 Now, we know that if the x,y coordinates of a point are integer, the sum will also be an integer
X + y = k (k, a variable) As per the question we need to exclude all the values lying on the boundary of triangle, k can take all values from 1 to 40 only. K = 0 is rejected as at k =0 will give the point at A which is also not allowed.
 With K = 40, x + y = 40; this will be satisfied by points (1, 39), (2, 38), (3,37) …… (38, 2), (39, 1). That is a total of 39 points
 Similarly x + y = 38, will be satisfied by 37 points.
X + Y = 37, will be satisfied by 36 points
X + Y = 3 will be satisfied by 2 points
X + Y = 2 will be satisfied by 1 point
X + Y = 1 will not be satisfied by any point So, the total number of all such points is: 39 + 38 + 37 + 36 + ……………………. + 3 + 2 + 1 = n(n+1)/2 points = (39*40) / 2 = 780 points
Example 5: Two lines P and Q intersect at point (3, 2) in the xy plane. The slope of line P is 45 degrees and line Q is parallel to the X axis. What is the area (in sq. units) of the triangle formed by P, Q and a line perpendicular to P passing through point (5, 4) ?
a) 12
b) 8
c) 6
d) 4
 Let us look at the image below:
 As slope of line P is 45 degree. Therefore, ∠ABC = 45 degree
 In triangle ABC, length of AB = SQRT [(53)^{2 }+ (42)^{2}] = 2√2 units
 Therefore, length of line AC = 2√2 units (Since ABC is an isosceles triangle. Thus AB = AC)
 Thus, required area = ½ * 2√2 * 2√2 = 4 sq. units
Example 6: The line √3 Y = x is the radius of the circle. It meets the circle o=centered at origin O at point M (√3, 1). If PQ is the tangent to the circle at M as shown, find the length of the PQ.
a) (5/2)√3 units
b) 3 √3 units
c) 2√3 units
d) 8/√3 units
 PQ is perpendicular to line Y = X / √3 (Since, radius of a circle is perpendicular to the tangent of the circle)
 Therefore, slope of PQ = 1 / (1/ √3) = – √3 (Since, product of slopes of line perpendicular to each other is 1)
 Therefore, Let equation of the line PQ be y = – √3x + c
 Now at the point M, when x = √3, y = 1
 Putting the above values of x and y in the above equation, we get c = 4 The equation of the line becomes, Y = – √3x + 4
 Thus, by using the above equation, we get:
 Coordinates of point P = (0, 4) and coordinates of point Q = (4/√3, 0) (Putting x = 0 in above equation, we find value of P and putting Y = 0 in above equation, we find value of Q)
 Hence PQ = sqrt [(4/√3) ^{2 }+ 4^{2}] = 8/√3 units.
Q1: What is the equation of a circle with centre of origin and radius is 6 cm?
Ans: (c)
Sol:
Given,
Center of the circle = (0,0)
Radius of the circle (r) = 6 cm
∴ Equation of the circle is
Hence, the correct answer is Option C
Q2: The equation of circle with centre (1, 2) and radius 4 cm is:
(a) x^{2} + y^{2 }+2x  4y = 11
(b) x^{2} + y^{2 }+ 2x  4y = 16
(c) x^{2} + y^{2}  2x + 4y = 16
(d) x^{2} + y^{2 } 2x + 4y = 11
Ans: (d)
Sol: Given,
Centre of the circle (a, b) = (1, 2)
Radius of the circle (r) = 4 cm
... Equation of the circle is (xa)^{2} + (y  b)^{2 }= r^{2}
=> (x1)^{2} +(y(2))^{2} = 4^{2}
=> (x1)^{2} + (y+2)^{2} = 4^{2}
=> x^{2}+1^{2 } 2.x.1 + y^{2} + 2^{2 }+2.y.2 = 16
=> x^{2 }+ 1  2x + y^{2 }+ 4 + 4y = 16
=> x^{2}  2x + y^{2} + 4y = 16  1  4
=> x^{2 }+ y^{2 } 2x + 4y = 11
Hence, the correct answer is Option (d)
Q3: In ΔABC, AB = AC. A circle drawn through B touches AC at D and intersect AB at P. If D is the mid point of AC and AP 2.5 cm, then AB is equal to:
(a) 9 cm
(b) 10 cm
(c) 7.5 cm
(d) 12.5 cm
Ans: (b)
Sol:
Given D is midpoint of AC so,
AD = AC/2
But also given AC = AB
AD = AB/2  (1)
AD is a tangent and APB is a secant. So the tangent secant theorem can be applied,
AD^{2} = AP X AB
(AB/4)^{2} = 2.5 × AB
AB^{2}/4 = 2.5 × AB
AB = 10 cm
181 videos146 docs111 tests

Coordinate Geometry  Finding the Slope of a Line Video  05:29 min 
Important Formulae: Coordinate Geometry Doc  2 pages 
1. What are the basic concepts of coordinate geometry? 
2. What are the formulas used in coordinate geometry? 
3. How do you find the equation of a line in coordinate geometry? 
4. What is the significance of knowledge of quadrants in coordinate geometry? 
5. Can you provide a solved example of coordinate geometry? 
181 videos146 docs111 tests

Coordinate Geometry  Finding the Slope of a Line Video  05:29 min 
Important Formulae: Coordinate Geometry Doc  2 pages 

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