Question

If a point $P(x,y)$ moves along the ellipse $\frac{x^2}{25}+\frac{y^2}{16}=1$ and if $C$ is the centre of the ellipse, then $4\text{max}\left(CP\right)+5\text{min}\left(CP\right)$ is equal to

• A. $25$
• B. $40$
• C. $45$
• D. $54$
• E. $16$

Answer 1

The correct option is B $40$

Given equation of ellipse is
$\frac{x^2}{5^2}+\frac{y^2}{4^2}=1$
$\Rightarrow a=6,b=4$ $[\because a>b]$
$CP$ is maximum when $P$ is at the end of major axis i.e. $\text{max}\left(CP\right)=5$. $CP$ is minimum when $P$ is at the end of minor axis i.e. $\text{min}\left(CP\right)=4$
Therefore, $4\text{max}\left(CP\right)+5\text{min}\left(CP\right)=4\times 5+5\times 4=40$