A tangent to the ellipse frac{x^2}{a^2} + fra | vedclass

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Question

(A) The equation of the tangent to the ellipse at $P(a \cos 	heta, b \sin 	heta)$ is given by $\frac{x}{a} \cos 	heta + \frac{y}{b} \sin 	heta = 1

Vedclass · Accepted Answer

$ab$

Vedclass · Answer

(A) The equation of the tangent to the ellipse at $P(a \cos 	heta, b \sin 	heta)$ is given by $\frac{x}{a} \cos 	heta + \frac{y}{b} \sin 	heta = 1$. Setting $y = 0$,we get the $x$-intercept $A = (\frac{a}{\cos 	heta}, 0)$. Setting $x = 0$,we get the $y$-intercept $B = (0, \frac{b}{\sin 	heta})$. The area of $\Delta OAB$ is given by $Area = \frac{1}{2} |x_A \cdot y_B| = \frac{1}{2} |\frac{a}{\cos 	heta} \cdot \frac{b}{\sin 	heta}| = \frac{ab}{|2 \sin 	heta \cos 	heta|} = \frac{ab}{|\sin 2	heta|}$. Since the minimum value of $|\sin 2	heta|$ is $1$ (when $2	heta = 90^\circ$ or $	heta = 45^\circ$),the minimum area is $ab$.

$A$ tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ at point $P$ meets the coordinate axes at points $A$ and $B$. Find the minimum area of $\Delta OAB$.

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