Let C be a circle having centre in the first quadrant and touching the x-axis at a distanc…

Question

Let $C$ be a circle having centre in the first quadrant and touching the $x$-axis at a distance of $3$ units from the origin. If the circle has an intercept of length $6\sqrt3$ on the $y$-axis, then the length of the chord of the circle $C$ on the line $x-y=3$ is:

Accepted Answer

Point of contact on the $x$-axis is $(3,0)$, so the centre is $(3,r)$ for radius $r$.

$y$-intercept: distance from centre to $y$-axis is $3$, so $2\sqrt{r^2-9}=6\sqrt3 \Rightarrow r^2-9=27\Rightarrow r=6$.

Centre $(3,6)$. Distance to $x-y=3$: $p=\dfrac{|3-6-3|}{\sqrt2}=\dfrac{6}{\sqrt2}=3\sqrt2$.

$$\text{Chord length}=2\sqrt{r^2-p^2}=2\sqrt{36-18}=2\sqrt{18}=6\sqrt2.$$

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