Conic Sections Question 370
Question: The locus of a variable point whose distance from (-2, 0) is $ \frac{2}{3} $ times its distance from the line $ x=-\frac{9}{2} $ , is
[IIT 1994]
Options:
A) Ellipse
B) Parabola
C) Hyperbola
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let point P $ (x_1,y_1) $
So, $ \sqrt{{{(x_1+2)}^{2}}+y_1^{2}}=\frac{2}{3}( x_1+\frac{9}{2} ) $
therefore $ {{(x_1+2)}^{2}}+y_1^{2}=\frac{4}{9}{{( x_1+\frac{9}{2} )}^{2}} $
therefore $ 9[x_1^{2}+y_1^{2}+4x_1+4]=4( x_1^{2}+\frac{81}{4}+9x_1 ) $
therefore $ 5x_1^{2}+9y_1^{2}=45 $
therefore $ \frac{x_1^{2}}{9}+\frac{y_1^{2}}{5}=1 $ , Locus of $ (x_1,y_1) $ is $ \frac{x^{2}}{9}+\frac{y^{2}}{5}=1 $ , which is equation of an ellipse.
BETA
