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If any tangent to the ellipse (x^2)/(a^2...

If any tangent to the ellipse `(x^2)/(a^2)+(y^2)/(b^2)=1` intercepts equal lengths `l` on the axes, then find `l`.

Text Solution

Verified by Experts

The correct Answer is:
`sqrt(a^(2)+b^(2))`
The equation of tangenet to the given ellipse at point `P(a cos theta, b sin theta)` is
`(x)/(a) cos theta+(y)/(b) sin theta=1`
The intercepts of line on the ellipse are `a//cos theta and b//sin theta`.
Given that
`(a)/(costheta)=(b)/(sin theta)l or cos theta=(theta)/(l)and sin theta=(b)/(l)`
or `cos^(2)theta+sin^(2)theta=(a^(2))/(l^(2))+(b^(2))/(l^(2))`
`or l^(2)=a^(2)+b^(2)`
or `l=sqrt(a^(2)+b^(2))`
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