Find the equations of the two circles which each satisfy the following conditions:

the axis of \(x\) is a tangent to the circle,

the centre of the circle lies on the line \(2y = x\),

the point \((14,2)\) lies on the circle.

The red circle here satisfies the first two conditions – it touches the \(x\)-axis, and its centre is on the blue line, \(2y = x\).

We can vary \(r\), the radius of the circle, so that the circle passes through \((14,2)\) – for how many values of \(r\) does this happen?

Prove that the line \(3y = 4x\) is a common tangent to these circles.

We could check how many intersections this line has with each of the circles.

Or, we could think about where this line meets the radius of the circle that is perpendicular to the line.

If we change the value of \(k\) in the applet above, the green line is \(y = kx\).