If any tangent to the ellipse frac{x^2}{a^2} | vedclass

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(C) The equation of the tangent to the ellipse at the point $(a \cos 	heta, b \sin 	heta)$ is given by $\frac{x \cos 	heta}{a} + \frac{y \sin 	het

Vedclass · Accepted Answer

$\frac{a^2}{h^2} + \frac{b^2}{k^2} = 1$

Vedclass · Answer

(C) The equation of the tangent to the ellipse at the point $(a \cos 	heta, b \sin 	heta)$ is given by $\frac{x \cos 	heta}{a} + \frac{y \sin 	heta}{b} = 1$.This can be rewritten in the intercept form $\frac{x}{a/\cos 	heta} + \frac{y}{b/\sin 	heta} = 1$.Comparing this with the intercept form $\frac{x}{h} + \frac{y}{k} = 1$,we get the intercepts $h = \frac{a}{\cos 	heta}$ and $k = \frac{b}{\sin 	heta}$.Therefore,$\frac{a}{h} = \cos 	heta$ and $\frac{b}{k} = \sin 	heta$.Squaring and adding these equations,we get $\cos^2 	heta + \sin^2 	heta = \frac{a^2}{h^2} + \frac{b^2}{k^2}$.Since $\cos^2 	heta + \sin^2 	heta = 1$,it follows that $\frac{a^2}{h^2} + \frac{b^2}{k^2} = 1$.

If any tangent to the ellipse $\frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ makes intercepts of length $h$ and $k$ on the axes,then:

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