Question
Let C be a circle having centre in the first quadrant and touching the x -axis at a distance 3 units from the origin. If the circle C has an intercept of length $6\sqrt{3}$ on $y$-axis, then the length of the chord of the circle $C$ on the line $x - y = 3$ is :
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Solution
Sol. [Key Concept: The centre is $(3,6)$ with radius $6$ (the $y$-intercept $6\sqrt{3}$ fixes it); the chord on $x - y = 3$ has length $2\sqrt{r^{2} - p^{2}}$ with $p$ the perpendicular distance.]
Centre of circle is $(3,6)$ and radius $= 6$ units.
$$\begin{matrix} & PQ = 2\sqrt{6^{2} - p^{2}}\ \therefore p = \frac{6}{\sqrt{2}} \\ & p = \frac{|3 - 6 - 3|}{\sqrt{2}} \\ & PQ = 2\sqrt{36 - \frac{36}{2}} \\ & \ = 2 \times \frac{6}{\sqrt{2}} = 6\sqrt{2} \end{matrix}$$
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