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

Answer: (D)

$9$

Answer 2

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

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