The minimum area of triangle formed by the tangent to the ellipse x 2 a 2 + y 2 b 2 = 1 and coordinate axes is [2005]

Question

The minimum area of triangle formed by the tangent to the ellipse $\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$ and coordinate axes is [2005]

Smart Achievers · Accepted Answer

(1)

$Any tangent to the ellipse \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1 at P (a \cos θ, b \sin θ) is$ $\frac{x \cos θ}{a} + \frac{y \sin θ}{b} = 1$

$It meets coordinate axes at A (a \sec θ, 0) and B (0, b c o s e c θ)$

$∴$ $Area of ∆ O A B = \frac{1}{2} \times a \sec θ \times b c o s e c θ = \frac{a b}{\sin 2 θ}$

$For area of ∆ O A B to be minimum, \sin 2 θ should be maximum, i.e., 1$

$∴$ $Minimum area of ∆ O A B = a b sq. units.$

Want Us to Email you About Special Offers & Updates?