Conic Sections Question 394
Question: A point ratio of whose distance from a fixed point and line $ x=9/2 $ is always 2 : 3. Then locus of the point will be
[DCE 2005]
Options:
A) Hyperbola
B) Ellipse
C) Parabola
D) Circle
Show Answer
Answer:
Correct Answer: B
Solution:
In question, $ PS=\frac{2}{3}PM $ (Given) Focus $ S(-2,0) $ , Equation of directrix $ 2x-9=0 $
$ {{(PS)}^{2}}=\frac{4}{9}{{(PM)}^{2}} $
therefore $ {{(h+2)}^{2}}+{{(k)}^{2}}=\frac{4}{9}{{( \frac{2h-9}{2} )}^{2}} $
therefore $ 9[{{(h+2)}^{2}}+{{(k)}^{2}}]=\frac{4{{(2h-9)}^{2}}}{4} $
therefore $ 9h^{2}+9k^{2}+36h+36=4h^{2}+81+36h $
therefore $ \frac{5h^{2}}{45}+\frac{9k^{2}}{45}=1 $
therefore $ \frac{h^{2}}{9}+\frac{k^{2}}{5}=1 $
therefore 1 Locus of point P(h, k) is $ \frac{x^{2}}{9}+\frac{y^{2}}{5}=1 $ , which is an ellipse.
BETA
