Question

A circle has a center that falls on the line `y = 7/9x +7` and passes through `( 4 ,1 )` and `(3 ,7 )`. What is the equation of the circle?

Answer 1

The circle is
`(x+129/22)^2+(y-161/66)^2=216413/2178`

Explanation:

The general equation of the circle, center `(a,b)`, radius `r` is
`(x-a)^2+(y-b)^2=r^2`.

Since the center lies on a given line, `b=7/9a+7`.
Since two given points line on the circle:
`(4-a)^2+(1-b)^2=r^2=(3-a)^2+(7-b)^2=r^2`
Hence `16-8a+cancel(a^2)+1-2b+cancel(b^2)=9-6a+cancel(a^2)+49=14b+cancel(b^2)`

So the problem reduces to solving for `a`, `b` the simultaneous equations above.

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