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  4. If the tangents on the ellipse 4x2 + y2 = 8 at the points (1,2) and (a, b) are perpendicular to each other, then a2 is equal to :

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Q. If the tangents on the ellipse $4x^2 + y^2 = 8$ at the points $(1,2)$ and $(a, b)$ are perpendicular to each other, then $a^2$ is equal to :

JEE MainJEE Main 2019Conic Sections

Solution:

$4a^{2 }+b^{2} = 8$ .......(1)
also $ \frac{dy}{dx}\Bigg)_{\left(1,2\right)} = - \frac{ 4x}{y} =-2$
$ \Rightarrow - \frac{4a}{b} = \frac{1}{2} b=-8a$
$ \Rightarrow b^{2} =64a^{2} 68a^{2}=8$
$ a^{2} = \frac{2}{17} $