Given three clues, can we find the equation of this circle?

Given three clues, can we find the equation of this circle?

Question

Find the equation of the circle which passes through the points $(2,0)$ and $(8,0)$ and touches the $y$-axis on the positive side of the origin.

Find the coordinates of the point on the circle which is furthest from the point $(2,0)$.

Suggestion

A sketch of the circle would help to visualise the problem.

Are there any circle theorems that could help you?

Solution

The vertical line of symmetry of the circle is $x = 5$, and so the centre of the circle must be a point of the form $(5,q)$.

As the circle touches the $y$-axis, the radius of the circle is therefore $5$.

The distance between $(2,0)$ and $(5,q)$ must be $5$, so $\sqrt{(2-5)^2 + q^2} = 5$. Solving for $q$ gives $q = 4$, and so the centre of the circle is $(5,4)$.

Therefore the equation of the circle is $(x - 5)^2 + (y - 4)^2 = 25.$

The point on the circle furthest from this point lies on the opposite end of the diameter that contains $(2,0)$.

By symmetry around the centre $(5,4)$, this point is $(8,8)$.