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Tangent is drawn to ellipse (x^2)/(27)+y...

Tangent is drawn to ellipse `(x^2)/(27)+y^2=1` at `(3sqrt(3)costheta,sintheta)` [where `theta in (0,pi/2)]` Then the value of `theta` such that sum of intercepts on axes made by this tangent is minimum is (a) `pi/3` (b) `pi/6` (c) `pi/8` (d) `pi/4`

Text Solution

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The equation of tangent at a given point on the ellipse is
`(x*sqrt(3)cos theta)/(27)+(y*sin theta)/(1)=1`
`or (x)/(3sqrt(3)sec theta)+(y)/("cosec"theta)=1`
Sun of the intercepts on axes is given by
`S=3sqrt(3)sectheta+"cosec" theta,theta in (o,(pi)/(2))`
To find the minimum value of S, we differentiates S w.r.t. ` theta`
`:. (dS)/(d theta)=3sqrt(3)sec thetatan theta-"cosec" thetacot theta`
If `(dS)/(d theta)=0`, then `tan^(3)=(1)/(sqrt(3))or tantheta=(1)/(sqrt(3))`
Therefore, `theta=(pi)/(6)`, for which value of S is minimum as `underset(thetararr0,(pi)/(2))lim(3sqrt(3)sec theta+"cosec"theta)rarroo`
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