Conic Sections Question 389
Question: Minimum area of the triangle by any tangent to the ellipse $ \frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 $ with the coordinate axes is
[IIT Screening 2005]
Options:
A) $ \frac{a^{2}+b^{2}}{2} $
B) $ \frac{{{(a+b)}^{2}}}{2} $
C) ab
D) $ \frac{{{(a-b)}^{2}}}{2} $
Show Answer
Answer:
Correct Answer: C
Solution:
Equation of tangent at $ (a\cos \theta ,\ b\sin \theta ) $ is $ \frac{x}{a}\cos \theta +\frac{y}{b}\sin \theta =1 $
$ P=( \frac{a}{\cos \theta },0 ) $
$ Q=( 0,\frac{b}{\sin \theta } ) $
Area of $ OPQ=\frac{1}{2}| ( \frac{a}{\cos \theta } )( \frac{b}{\sin \theta } ) |=\frac{ab}{|\sin 2\theta |} $
$ {{(Area)} _{\min }}=ab $ .
BETA
