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 $
$\Rightarrow e ^2=1-\frac{3}{4}=\frac{1}{4} \Rightarrow e =\frac{1}{2}$