Conic Sections Question 490
Question: If the normal at any point P on the ellipse cuts the major and minor axes in G and g respectively and C be the centre of the ellipse, then
[Kurukshetra CEE 1998]
Options:
A) $ a^{2}{{(CG)}^{2}}+b^{2}{{(Cg)}^{2}}={{(a^{2}-b^{2})}^{2}} $
B) $ a^{2}{{(CG)}^{2}}-b^{2}{{(Cg)}^{2}}={{(a^{2}-b^{2})}^{2}} $
C) $ a^{2}{{(CG)}^{2}}-b^{2}{{(Cg)}^{2}}={{(a^{2}+b^{2})}^{2}} $
D) None of these
Show Answer
Answer:
Correct Answer: A
Solution:
Let at a point $ (x_1,y_1) $ normal will be $ \frac{(x-x_1)a^{2}}{x_1}=\frac{(y-y_1)b^{2}}{y_1} $
At $ G,y=0 $
therefore $ x=CG=\frac{x_1(a^{2}-b^{2})}{a^{2}} $
At $ g,x=0 $
therefore $ y=Cg=\frac{y_1(b^{2}-a^{2})}{b^{2}} $
$ \frac{x_1^{2}}{a^{2}}+\frac{y_1^{2}}{b^{2}}=1 $
therefore $ a^{2}{{(CG)}^{2}}+b^{2}{{(Cg)}^{2}}={{(a^{2}-b^{2})}^{2}}. $
BETA
