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 + x² - 4x = 4 

2x² - 4x + 4 - 4  =  0

2x² - 4x  =  0

2x (x - 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

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

Related pages

• How to find the length of tangent
• How to find equation of tangent at the given point
• Inside outside or on the circle

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