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Find the equation of the circle passing ...

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

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Let equation of circle be
`x^(2)+y^(2)+2gx+2fy+c=0`
This circle passes through the point (4,1).
`:.4^(2)+1^(2)+2(4)+2f(1)+c=0`
`rArr" "8g+2f+c=-17` . . . (1)
Given circle passes through the point (6,5).
`:.6^(2)+5^(2)+2g(6)+2f(5)+c=0`
`rArr" "12g+10f+c=-61` . . . (2)
The centre (-g,-f) of circle lies on the line
4x+y=16
`:.` -4g-f=16 . . . (3)
Subtracting equation (1) from equation (2),
4g+8f=-44 . . . (4)
Adding equations (3) and (4),
7f=-28
`rArr" "f=-4`
Put the value of f in equation (3)
-4g+4=16
`rArr" "-4g=12`
`rArr" "-4g=12`
`rArr" "g=-3`
Put the of g and f in equation (1)
-24-8+c-17
`rArr" "c=15`
Therefore equation of circle is
`x^(2)+y^(2)-6x-8y+15=0`
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