Find the equation of a circle with centre (2, 2) and passes through the point (4, 5).
Given the center of the circle as ( 2 , 2 ) and the point through which the circle passes is ( 4 , 5 )
The equation of the required circle is given by,
( x − h ) 2 + ( y − k ) 2 = r 2 (1)
Where h and k denotes the center of the circle and r denotes the radius of the circle.
As the circle passes through the point ( 4 , 5 ) , then the radius of the circle is the distance between the points ( 2 , 2 ) and ( 4 , 5 )
The distance between two points ( x 1 , x 2 ) and ( y 1 , y 2 ) is given by the formula,
( x 1 − x 2 ) 2 + ( y 1 − y 2 ) 2 (2)
Substitute the values of x 1 , x 2 , y 1 , y 2 in equation (2) to determine the radius of the circle.
r = ( 2 − 4 ) 2 + ( 2 − 5 ) 2 ( − 2 ) 2 + ( − 3 ) 2 4 + 9 13
Substitute the values of h = 2 , k = 2 , r = 13 in equation (1)
( x − h ) 2 + ( y − k ) 2 = r 2
( x − 2 ) 2 + ( y − 2 ) 2 = ( 13 ) 2 x 2 − 4 x + 4 + y 2 − 4 y + 4 = 13 x 2 − 4 x + y 2 − 4 y + − 5 = 0
Thus the equation of the circle with center (2,2) and which passes through (4,5) is x 2 − 4 x + y 2 − 4 y + − 5 = 0
Given the center of the circle as
The equation of the required circle is given by,
Where h and k denotes the center of the circle and r denotes the radius of the circle.
As the circle passes through the point
The distance between two points
Substitute the values of
Substitute the values of
Thus the equation of the circle with center (2,2) and which passes through (4,5) is
