Question

The locus of the mid-point of the portion intercepted between the axes by the line $x\cos \alpha +y\sin \alpha =p$ passes through the point $(p+1,p-1)$, then $71(4p^4-20p^2+99)$ is equal to

Answer 1

Mid-point of the portion intercepted between the axes by the line $x\cos \alpha +y\sin \alpha =p$ is $(\frac{p}{2\cos \alpha },\frac{p}{2\sin \alpha })$

$\Rightarrow \frac{p}{2\cos \alpha }=h$ and $\frac{p}{2\sin \alpha }=k$ (Let)

$\Rightarrow \cos \alpha =\frac{p}{2h}$ and $\sin \alpha =\frac{p}{2k}$

Now, Square both equation and add,

$\Rightarrow \frac{p^2}{4h^2}+\frac{p^2}{4k^2}=1$ (Locus)

Now, replace $(h,k)$ with $(p+1,p-1)$

$\Rightarrow \frac{p^2}{4{(p+1)}^2}+\frac{p^2}{4{(p-1)}^2}=1$

$\Rightarrow 2\left[{(p^2-1)}^2\right]=p^2(p^2+1-2p)+p^2(p^2+1+2p)$

$\Rightarrow p^2=\frac{1}{3}$

So, $71(4p^4-20p^2+99)=71\times 91=6461$