First slide

Ellipse

Question

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

Moderate

A

$a b sq. units$
B

$\frac{a^{2} + b^{2}}{2} sq. units$
C

$\frac{(a + b)^{2}}{2} sq. units$
D

$\frac{a^{2} + a b + b^{2}}{3} sq. units$

Solution

Any tangent to the ellipse
$\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1$

$at P (a \cos θ, b \sin θ) is$
given by
$\frac{x \cos θ}{a} + \frac{y \sin θ}{b} = 1$
$It meets the coordinate axes at A (a \sec θ, 0) and B (0, b cosec θ) . Therefore,$
$Area of Δ O A B = \frac{1}{2} \times a \sec θ \times b cosec θ$
$or Δ = \frac{a b}{\sin 2 θ}$
$For area to be minimum, \sin 2 θ should be maximum and we$
$know that the maximum value of \sin 2 θ is 1 . Therefore,$
$Δ_{max} = a b$

Get Instant Solutions

When in doubt download our app. Now available Google Play Store- Doubts App

Recieve an sms with download link

Doubts App

OR

SCAN ME

Doubts App

Doubts App