Mathematics - Ellipse Question with Solution | TestHub

Let P be $(5\cos \theta ,4\sin \theta );\mathrm{Q}$ be $(-5\sin \theta ,4\cos \theta )$

Equation of tangent at $\mathrm{P}\frac{\mathrm{x}}{5}\cos \theta +\frac{\mathrm{y}}{4}\sin \theta =1$ .............(i)

Equation of tangent at $\mathrm{Q}-\frac{\mathrm{x}}{5}\sin \theta +\frac{\mathrm{y}}{4}\cos \theta =1$ .........(ii)

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 $\Rightarrow \mathrm{m}:\mathrm{n}=1:1\Rightarrow \mathrm{m}+\mathrm{n}=2$