Finding the equation of the circle given two points and a line passing through the center using the Perpendicular Bisector Theorem

Question

So this is the problem:

Determine the equation of the circle passing through $(4,0)$ and $(3,5)$ with a line $3x+2y-7=0$ passing through the center.

I know the easier solution of this, but my teacher wants us to solve using our knowledge in Perpendicular Bisector Theorem. So can anyone help me?

Answer 1

HINT

The center has coordinates

• $C=(t,(7-3t)/2)$

from the condition

$$d^2(CA)=d^2(CB) \implies (t-4)^2+\left(\frac{7-3t}{2}-0\right)^2=(t-3)^2+\left(\frac{7-3t}{2}-5\right)^2$$

we can find $t$.

Answer 2

The chord of the circle connecting $(4,0)$ and $(3,5)$ has slope $-5$. The perpendicular bisector of the chord will be a radius: this line passes through the mid-point of chord, $(7/2, 5/2)$ and has slope $1/5$. Its intersection with $3x+2y-7=0$ will be the centre of the circle.