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Conic Sections — Maths STD 11 Science — Question

Maharashtra BoardEnglish MediumSTD 11 ScienceMathsConic Sections2 Marks
MCQ
The locus of a variable point whose distance from (-2,0) is $\frac{2}{3}$ times its distance from the $\operatorname{linc} x=-\frac{9}{2}$, is
  • ✓
    Ellipse
  • B
    Parabola
  • C
    Hyperbola
  • D
    None of these
✓

Answer

Correct option: A.
Ellipse
(A)
Let point $P \left(x_1, y_1\right)$
$\begin{array}{l}\therefore \sqrt{\left(x_1+2\right)^2+y_1^2}=\frac{2}{3}\left(x_1+\frac{9}{2}\right)
\\ \Rightarrow\left(x_1+2\right)^2+y_1^2=\frac{4}{9}\left(x_1+\frac{9}{2}\right)^2
\\ \Rightarrow 9\left(x_1^2+y_1^2+4 x_1+4\right)=4\left(x_1^2+\frac{81}{4}+9 x_1\right) \\ \Rightarrow 5 x_1^2+9 y_1^2=45
\\ \Rightarrow \frac{x_1^2}{9}+\frac{y_1^2}{5}=1\end{array}$
∴ Locus of $\left(x_1, y_1\right)$ is $\frac{x^2}{9}+\frac{y^2}{5}=1$, which is equation of an ellipse.

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