Question

If the equation of the ellipse passing through $(2,1)$ and having $e=\frac{1}{2}$ is $p{x}^2+q{y}^2=16$. Find the value of $p+q$.

• A. $5$
• B. $6$
• C. $7$
• D. $8$

Answer 1

Answer: (C)

$7$

Let the ellipse be $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1.$
Given it passes through $(2,1)$
$\therefore \frac{4}{a^2}+\frac{1}{b^2}=1$ ...(1)
or $\frac{4}{a^2}+\frac{1}{a^2(1-1/4)}=1$ $(\because e=\frac{1}{2})$
or $4+\frac{4}{3}=a^2$ or $a^2=\frac{16}{3}$
$\therefore \frac{1}{b^2}=1-\frac{4}{a^2}=1-\frac{3}{4}=\frac{1}{4}$ from (1)
Hence required ellipse is, $\frac{3x^2}{16}+\frac{y^2}{4}=1$ or $3x^2+4y^2=16$