Question

A point $P$ moves such that the sum of twice its distance from the origin and its distance from the $y$-axis is a constant equal to 3. P describes

• A. a circle with its centre at $(-1,0)$ and radius $2\sqrt{3}$.
• B. an ellipse centred at $(-1,0)$ and of eccentricity $\frac{1}{2}$
• C. a hyperbola centred at (1, 0) and of eccentricity 2
• D. none of these

Answer 1

Answer: (B)

Let $P(h,k)$
$\Rightarrow 2\sqrt{h^2+k^2}+|h|=3$
$\Rightarrow 4\left(h^2+k^2\right)=9+h^2-6|h|$
$\Rightarrow \frac{(|h|+1{)}^2}{4}+\frac{k^2}{3}=1$
$\Rightarrow e^2=1-\frac{3}{4}=\frac{1}{4}\Rightarrow e=\frac{1}{2}$