Coordinate Geometry Notes for CUET, CLAT & IPMAT
Coordinate Geometry is one of the most important topics in Quantitative Aptitude. Questions are frequently asked in entrance exams like CUET , CLAT , and IPMAT . This chapter combines Algebra + Geometry and focuses on points, lines, distance, slope, and circles on a coordinate plane.
1. Introduction to Coordinate Geometry Coordinate Geometry is the study of geometry using numbers and equations.
It was developed by René Descartes
In coordinate geometry, every point is represented on the Cartesian Plane using coordinates.
2. Cartesian Coordinate System A coordinate plane consists of:
X-axis → Horizontal line
Y-axis → Vertical line
Origin (0,0) → Point where both axes meet
The plane is divided into 4 quadrants .
Signs of Coordinates in Quadrants
Quadrant Sign of x Sign of y I + + II - + III - - IV + -
Example
Point (3,4) lies in First Quadrant because both coordinates are positive.
Point (-2,5) lies in Second Quadrant.
3. Distance Formula The distance between two points:
A ( x 1 , y 1 ) , B ( x 2 , y 2
is given by:
d= sqrt{(x_2-x_1)^2 + (y_2-y_1)^2}
Explanation This formula is derived from the Pythagoras Theorem.
Example 1 Find distance between (2,3) and (6,7)
d = ( 6 − 2 ) 2 + ( 7 − 3 ) 2 d=\sqrt{(6-2)^2+(7-3)^2} d = ( 6 − 2 ) 2 + ( 7 − 3 ) 2 = 4 2 + 4 2 =\sqrt{4^2+4^2} = 4 2 + 4 2 = 16 + 16 =\sqrt{16+16} = 16 + 16 = 32 = 4 2 =\sqrt{32}=4\sqrt2
Shortcut If points lie on:
Same x-axis → distance = difference of x coordinates
Same y-axis → distance = difference of y coordinates
4. Section Formula Used to find a point dividing a line segment in a given ratio.
If point P divides A(x₁,y₁) and B(x₂,y₂) in ratio m:n internally:
P ( m x 2 + n x 1 m + n , m y 2 + n y 1 m + n ) P\left(\frac{mx_2+nx_1}{m+n},\frac{my_2+ny_1}{m+n}\right) P ( m + n m x 2 + n x 1 , m + n m y 2 + n y 1 )
Example Find point dividing (1,2) and (5,6) in ratio 1:1.
P = ( 1 ( 5 ) + 1 ( 1 ) 2 , 1 ( 6 ) + 1 ( 2 ) 2 ) P=\left(\frac{1(5)+1(1)}{2},\frac{1(6)+1(2)}{2}\right) P = ( 2 1 ( 5 ) + 1 ( 1 ) , 2 1 ( 6 ) + 1 ( 2 ) ) = ( 3 , 4 ) =(3,4) = ( 3 , 4 )
5. Midpoint Formula Midpoint means the exact center between two points.
Formula:
M ( x 1 + x 2 2 , y 1 + y 2 2 ) M\left(\frac{x_1+x_2}{2},\frac{y_1+y_2}{2}\right)
Example
Find midpoint of (2,4) and (8,10)
M = ( 2 + 8 2 , 4 + 10 2 ) M=\left(\frac{2+8}{2},\frac{4+10}{2}\right) M = ( 2 2 + 8 , 2 4 + 10 ) = ( 5 , 7 ) =(5,7) = ( 5 , 7 )
6. Centroid of Triangle Centroid is the intersection point of medians.
For vertices:
( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) (x_1,y_1),(x_2,y_2),(x_3,y_3) ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) Centroid:
( x 1 + x 2 + x 3 3 , y 1 + y 2 + y 3 3 ) \left(\frac{x_1+x_2+x_3}{3},\frac{y_1+y_2+y_3}{3}\right) ( 3 x 1 + x 2 + x 3 , 3 y 1 + y 2 + y 3 )
Example
Vertices:
( 1 + 4 + 7 3 , 2 + 5 + 8 3 ) = ( 4 , 5 )
7. Area of Triangle For three points:
( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) (x_1,y_1),(x_2,y_2),(x_3,y_3) ( x 1 , y 1 ) , ( x 2 , y 2 ) , ( x 3 , y 3 ) Area formula:
Area = 1 2 ∣ x 1 ( y 2 − y 3 ) + x 2 ( y 3 − y 1 ) + x 3 ( y 1 − y 2 ) ∣ \text{Area}=\frac12\left|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)\right| Area = 2 1 ∣ x 1 ( y 2 − y 3 ) + x 2 ( y 3 − y 1 ) + x 3 ( y 1 − y 2 ) ∣
Important Concept
If area = 0, then points are collinear .
Example
Find area of triangle with vertices:
Area = 1 2 [ 1 ( 1 − 5 ) + 4 ( 5 − 1 ) + 4 ( 1 − 1 ) ] = 1 2 [ − 4 + 16 + 0 ] = 6
8. Slope of a Line Slope represents steepness of a line.
Formula:
m = y 2 − y 1 x 2 − x 1 m=\frac{y_2-y_1}{x_2-x_1}
Interpretation
Slope Meaning Positive Rising line Negative Falling line Zero Horizontal line Undefined Vertical line
Example
Find slope between (2,3) and (6,11)
9. Parallel and Perpendicular Lines Parallel Lines Two lines are parallel if:
m 1 = m 2 m_1=m_2 m 1 = m 2
Perpendicular Lines Two lines are perpendicular if:
m 1 m 2 = − 1 m_1m_2=-1 m 1 m 2 = − 1
Example Slope of first line = 2
Slope of perpendicular line:
− 1 2 -\frac12
10. Equation of Line
(a) Slope Intercept Form
y = m x + c
Where:
m = slope
c = y-intercept
Example
y = 2 x + 3
Slope = 2
Y-intercept = 3
1. Distance between (1,2) and (4,6)
A) 3
Answer: C) 5
2. Midpoint of (2,4) and (6,8)
A) (2,6)
Answer: B) (4,6)
3. Slope of line joining (1,3) and (5,11)
A) 1
Answer: B) 2
4. Which point lies in third quadrant?
A) (2,3)
Answer: D) (-2,-3)
5. Equation of x-axis
A) x = 0
Answer: B) y = 0
6. Equation of y-axis
A) x = 0
Answer: A) x = 0
7. If slope is zero, line is:
A) Vertical
Answer: B) Horizontal
8. If slope is undefined, line is:
A) Horizontal
Answer: B) Vertical
9. Distance of point (3,4) from origin
A) 3
Answer: C) 5
10. For parallel lines:
A) Slopes opposite
Answer: B) Slopes equal
11. For perpendicular lines:
A) m 1 = m 2 m_1=m_2 m 1 = m 2 m 1 + m 2 = 0 m_1+m_2=0 m 1 + m 2 = 0 m 1 m 2 = − 1 m_1m_2=-1 m 1 m 2 = − 1 m 1 m 2 = 1 m_1m_2=1 m 1 m 2 = 1
Answer: C) m 1 m 2 = − 1 m_1m_2=-1 m 1 m 2 = − 1
12. Equation of line with slope 3 and intercept 2
A) y = 2 x + 3 y=2x+3 y = 2 x + 3 y = 3 x + 2 y=3x+2 y = 3 x + 2 y = 3 x − 2 y=3x-2 y = 3 x − 2 y = 2 x − 3 y=2x-3 y = 2 x − 3
Answer: B) y = 3 x + 2 y=3x+2 y = 3 x + 2
13. Area of triangle formed by (0,0), (4,0), (0,3)
A) 6
Answer: A) 6
14. Center of circle:
( x − 2 ) 2 + ( y − 3 ) 2 = 25 (x-2)^2+(y-3)^2=25 ( x − 2 ) 2 + ( y − 3 ) 2 = 25
A) (2,3)
Answer: A) (2,3)
15. Radius of circle:
x 2 + y 2 = 49 x^2+y^2=49 x 2 + y 2 = 49
A) 49
Answer: C) 7
16. Slope of line parallel to:
y = 4 x + 5 y=4x+5 y = 4 x + 5
A) 51 4 \frac14 4 1
Answer: B) 4
17. Slope of line perpendicular to:
y = 2 x + 1 y=2x+1 y = 2 x + 1
A) 2− 1 2 -\frac12 − 2 1 1 2 \frac12 2 1
Answer: C) − 1 2 -\frac12 − 2 1
18. Point (0,0) is called:
A) Centroid
Answer: C) Origin
19. Area formed by three collinear points:
A) 1
Answer: D) 0
20. Equation of circle centered at origin with radius 5
A) x 2 + y 2 = 25 x^2+y^2=25 x 2 + y 2 = 25 x 2 + y 2 = 5 x^2+y^2=5 x 2 + y 2 = 5 x + y = 25 x+y=25 x + y = 25 x 2 − y 2 = 25 x^2-y^2=25 x 2 − y 2 = 25
Answer: A) x 2 + y 2 = 25 x^2+y^2=25 x 2 + y 2 = 25
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