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`.

Options

A. `a^(2) + b^(2)`
B. `sqrt(a^(2) + b^(2))`
C. `(a^(2) + b^(2))^(2)`
D. none of these

Answer

B

Explanation

The equation of any tangent to the given ellipse is `x/a costheta + y/b sin theta = 1`
This line meets the coordinate axes at
`P((a)/(cos theta),0) and Q (0, (b)/(sin theta))`
`therefore (a)/(cos theta) = l = (b)/(sin theta)`
`rArr cos theta = a/l and sin theta = b/l`
`rArr cos^(2)theta + sin^(2)theta = a^(2)/l^(2) + b^(2)/l^(2) rArr l^(2) = a^(2) + b^(2) rArr l = sqrt(a^(2) + b^(2))`