Q57 / 110 · VITEEE 2023 · Mathematics
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
- AA circle with its centre at \((-1,0)\) and radius \(2 \sqrt{3}\)
- Correct answerBAn ellipse centred at \((-1,0)\) and of eccentricity \(\frac{1}{2}\)
- CA hyperbola centred at \((1,0)\) and of eccentricity 2
- DNone of these
Solution
\(\begin{aligned}
& 2 \sqrt{\mathrm{x}^2+\mathrm{y}^2}+\mathrm{x}=3 \\
& \Rightarrow \frac{(\mathrm{x}+1)^2}{4}+\frac{\mathrm{y}^2}{3}=1 .
\end{aligned}\)
This represents an ellipse centred at \((-1,0)\) and
\(\text { of eccentricity }=\sqrt{\frac{4-3}{4}}=\frac{1}{2} \text {. }\)
Hence, the correct answer is option: 'B'.