Solving Equation of Circle

Below are the set of problems which explains how to solve equation of circles. Equation of circle can be determine on point of intersection. origin, at some point on coordinate system with x and y values.

Example 1: Find the equation of the circle whose center is the point of intersection of the lines 2x-3y=1 and 3x+4y=10 and passes through the origin.

Solution :

The point of intersection of two lines is (2, 1).
Therefore the center is (2, 1). Since the origin lies on the circle,
We get r= √[(2-0)²+ (1-0)²] =√5.
The equation becomes (x-2)²+(y-1)²=5.

Example 2: If the parametric form of the circle is given by x = - 4 + 5 Cosθ and
y = - 3 + 5 Sinθ. Find its equation in the Cartesian form.

Solution :

x = - 4 + 5 cosθ and y = - 3 + 5 Sinθ.
=> x + 4 = 5cosθ and y + 3 = 5 Sin θ.
Squaring and adding we have (x+4)²+ (y+3)²= 25.
So, the equation of the circle is (x+4)²+ (y+3)²= 25.

Example 3: The sides of a square are x=2, x=3, y=1 and y=2. Find the the equation of the circle drawn on the diagonal of the square as diameter.

Solution :

Let ABCD be a square and the equations of the sides AB, BC,CD and DA are y=1, x=3, y=2 and x=2. respectively.
Then A =(2,1) , B= (3,1) , C= (3,2) and Since the diagonals are diameters of the circle,the equation of the circle is
(x-2)(x-3)+(y-1)(y-2)=0
=> x²+y²-5x-3y+8=0 .