# If the point P on the curve,

Question:

If the point $P$ on the curve, $4 x^{2}+5 y^{2}=20$ is farthest from the point $\mathrm{Q}(0,-4)$, then $\mathrm{PQ}^{2}$ is equals to :

1. (1) 36

2. (2) 48

3. (3) 21

4. (4) 29

Correct Option: 1,

Solution:

Ellipse $\equiv \frac{x^{2}}{5}+\frac{y^{2}}{4}=1$

Let a point on ellipse be $(\sqrt{5} \cos \theta, 2 \sin \theta)$

$\therefore P Q^{2}=(\sqrt{5} \cos \theta)^{2}+(-4-2 \sin \theta)^{2}$

$=5 \cos ^{2} \theta+4 \sin ^{2} \theta+16+16 \sin \theta$

$=21+16 \sin \theta-\sin ^{2} \theta$

$=21+64-(\sin \theta-8)^{2}=85-(\sin \theta-8)^{2}$

$P Q^{2}$ to be maximum when $\sin \theta=1$

$\therefore P Q_{\max }^{2}=85-49=36$