Two chords are drawn from the point $P(h,k)$ on the circle $x^{2} + y^{2} = hx + ky$. If the $y$-axi

Question

Two chords are drawn from the point $P(h,k)$ on the circle $x^{2} + y^{2} = hx + ky$. If the $y$-axis divides both the chords in the ratio $2:3$, then which of the following may be correct?

Accepted Answer

[Key Concept: Put the $y$-axis section ($x = - \frac{2h}{3}$ and $x = - \frac{3h}{2}$) into the circle; real chords ($D > 0$) give the inequalities in $h,k$.]Put $x = \frac{- 2h}{3},y^{2} - ky + \frac{10h^{2}}{9} = 0$$$D > 0 \Rightarrow k^{2} - \frac{40}{9}h^{2} > 0 $$put $x = \frac{- 3h}{2}$$$\Rightarrow y^{2} - ky + \frac{15h^{2}}{4} = 0$$$$\begin{matrix} D & \ > 0 \Rightarrow k^{2} - 15h^{2} > 0 \ k^{2} & \ > 15h^{2} \end{matrix}$$

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