Question

# Find the equation of a circle with centre (2, 2) and passes through the point (4, 5).

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Solution

## 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 −4x+4+ y 2 −4y+4=13 x 2 −4x+ y 2 −4y+−5=0 Thus the equation of the circle with center (2,2) and which passes through (4,5) is x 2 −4x+ y 2 −4y+−5=0

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