Mathematics - Parabola Question with Solution | TestHub

Let $\mathrm{P},\mathrm{Q}$ be two points on the ellipse $\frac{{\mathrm{x}}^2}{25}+\frac{{\mathrm{y}}^2}{16}=1$ whose eccentric angles differ by a right angle. Tangents are drawn at $P$ and $Q$ to meet at $R$. If the chord $PQ$ divides the joint of $C$ and $R$ in the ratio $m:n$ ( $C$ being centre of ellipse), then find $m+n(m:n$ is in simplified form). Ans: 2 Sol: Let P be $(5\cos \theta ,4\sin \theta );\mathrm{Q}$ be $(-5\sin \theta ,4\cos \theta )$ Equation of tangent at $\mathrm{P}\frac{\mathrm{x}}{5}\cos \theta +\frac{\mathrm{y}}{4}\sin \theta =1$ Equation of tangent at $\mathrm{Q}-\frac{\mathrm{x}}{5}\sin \theta +\frac{\mathrm{y}}{4}\cos \theta =1$ Solving (i) and (ii) $\Rightarrow \mathrm{R}=(5(\cos \theta -\sin \theta ),4(\sin \theta +\cos \theta ))$ $\therefore \mathrm{m}:\mathrm{n}$ is $1:1$ $\Rightarrow \mathrm{m}+\mathrm{n}=2$ Alternate : Let $\mathrm{P}(5,0),\mathrm{Q}(0,4)$ $\Rightarrow \mathrm{R}(5,4)$ Intersection of CR and PQ is $\left(\frac{5}{2},2\right)$, which is mid poi8nt of CR $\Rightarrow \mathrm{m}:\mathrm{n}=1:1\Rightarrow \mathrm{m}+\mathrm{n}=2$