## About "How to find the point of intersection of circle and line"

How to find the point of intersection of circle and line ?

Here we are going to see how to find the point of intersection of circle and line.

To find the point (or) points of intersection of any two curves, we have to solve the equations and find the values of x and y.

Let us look into an example problem to understand the above concept.

Example 1 :

Find the co-ordinates of the point of intersection of the line x + y = 2 with the circle x2 + y2 = 4

Solution :

Equation of the given line

x + y  =  2

y =  2 - x  ----(1)

Equation of circle

x2 + y2 = 4  -----(2)

Applying the value of y in the second equation, we get

x2 + (2 - x)2 = 4

x2 + 22 + x2 - 2(2) x = 4

x2 + 4 + x- 4x = 4

2x2 - 4x + 4 - 4  =  0

2x2 - 4x  =  0

2x (x - 2)  =  0

 2x  =   0 x = 0  y =  2 - x y =  2 - 0y  =  2(0, 2) x - 2  =  0x  =  2y =  2 - 2 y =  0(2, 0)

Hence the points of intersection of the given circle and the line are (0, 2) and (2, 0).

Example 2 :

Find the co-ordinates of the point of intersection of the line x − 2y = 1 and the circle x2 + y2 − 2x − y + 1 = 0

Solution :

Equation of the line :

x − 2y = 1

x  =  1 + 2y  --------(1)

Equation of circle :

x2 + y2 − 2x − y + 1 = 0   --------(2)

Applying the value of x in the second equation, we get

(2y + 1)2 + y2 − 2(2y + 1) − y + 1 = 0

4y2 + 4y + 1 + y2 − 4y − 2 − y + 1 = 0

5y2 − y  = 0

y (5y - 1)  =  0

 y = 0x  =  1 + 2y  x  =  1 + 2(0)x  =  1 5y - 1  =  0y  =  1/5x  =  1 + 2y  x  =  1 + 2(1/5)  =  1 + (2/5)  =  7/5

Hence the points of intersection are (1, 0) and (7/5, 1/5).

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