Question

A circle with centre in the first quadrant is tangent to $y=x+10,y=x-6$ and the y-axis. Let $(h,k)$ be the centre of the circle. If the value of $(h+k)=a+b\sqrt{a}$ where $\sqrt{a}$ is a surd, find the value of $a+b$

Answer 1

$y=x+10$ and $y=x-6$ have same slope.

ie, for these lines to be the tangent to the circle, the perpendicular distance between them should be equal to the diametre of the circle.

And the horizontal length between them$=10-(-6)=16$

ie, $y=x-6+8=x+2$ passes through the centre of the circle.

$\Rightarrow k=h+2$

So, radius of the circle$=\frac{1}{2}|\left(16i\right)\times \frac{(i+j)}{\sqrt{2}}|=4\sqrt{2}$

As y-axis is a tangent to the circle, $h=4\sqrt{2}$

ie, $k=2+4\sqrt{2}$

So, $a+b=2+8=10$