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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,

( xh ) 2 + ( yk ) 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= ( 24 ) 2 + ( 25 ) 2 ( 2 ) 2 + ( 3 ) 2 4+9 13

Substitute the values of h=2,k=2,r= 13 in equation (1)

( xh ) 2 + ( yk ) 2 = r 2

( x2 ) 2 + ( y2 ) 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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