Question

Find the equation of the circle passing through the points (4, 1) and (6, 5) and whose centre is on the line 4 x + y = 16.

Answer 1

The given points through which circle passes are ( 4,1 ) and ( 6,5 ) and the line on which center lies is 4x+y=16 .

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.

Since circle passes through the point ( 4,1 ) , this point will satisfy the equation of the circle represented in equation ( 1 ) .

( 4−h ) 2 + ( 1−k ) 2 = r 2 (2)

Since circle also passes through the point ( 6,5 ) , this point will also satisfy the equation of the circle represented in equation ( 1 )

( 6−h ) 2 + ( 5−k ) 2 = r 2 (3)

Equate equation ( 2 ) and ( 3 ) to determine the relation between h and k ..

16+ h 2 −8h+1+ k 2 −2k=36+ h 2 −12h+25+ k 2 −10k 16−8h+1−2k=36−12h+25−10k 4h+8k=44 h+2k=11 (4)

Now since center lies on 4x+y=16 , Center ( h,k ) will satisfy this equation.

So, 4h+k=16 (5)

Solve equation ( 4 ) and ( 5 ) to determine the value of h and k ,

Multiply equation ( 4 ) by 4 and subtract from equation ( 5 ) we get,

4h+8k−44−( 4h−k−16 )=0 7k−28=0 7k=28 k=4

Substitute the value of k in equation ( 5 ) ,

4h+4=16 4h=12 h= 12 4 h=3

Substitute the value of h and k in equation (1) to obtain the value of r 2 .

( 4−3 ) 2 + ( 1−4 ) 2 = r 2 ( 1 ) 2 + ( −3 ) 2 = r 2 1+9= r 2 10= r 2

Substitute the value of h and k and r 2 in equation (1) to obtain the equation of circle.

( x−3 ) 2 + ( y−4 ) 2 =10

Thus the equation of the circle passing through ( 4,1 ) and ( 6,5 ) and whose center is on the line 4x+y−16=0 is ( x−3 ) 2 + ( y−4 ) 2 =10 .