Question

If the tangent at a point on the ellipse $\frac{x^2}{27} + \frac{y^2}{3} =1$ meets the coordinate axes at A and B, and O is the origin, them the minimum area (in s units) of the triangle OAB is:

• A. $\frac{9}{2}$
• B. $3 \sqrt{3}$
• C. $9 \sqrt{3}$
• D. $9$

Answer 1

The Correct Option is D

Equation of tangent to ellipse 
\(\frac{x}{\sqrt{27}}+\frac{y}{\sqrt{3}}=1\) 
Area bounded by line and co-ordinate axis 

\(\frac12\times\)intercept on x-axis \(\times\) intercept on y -axis
\(\Delta=\frac{1}{2}.\frac{\sqrt{27m^2+3}}{m}. {\sqrt{27m^2+3}}{sin}\)

\(\frac12\times\frac{(27m^2+3)}{m}\)

now apply 

AM≥GM

\(\frac{27m+\frac3m}{2}\)≥\(\sqrt{27m\times\frac3m}\)  ≥ \(9\)
\(\Delta\)
\(\Delta_{min}=9\)