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If the tangent at any point of the ellip...

If the tangent at any point of the ellipse `(x^2)/(a^3)+(y^2)/(b^2)=1` makes an angle `alpha` with the major axis and an angle `beta` with the focal radius of the point of contact, then show that the eccentricity of the ellipse is given by `e=cosbeta/(cosalpha)`

Text Solution

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Let`P(a cos theta, b sin theta)` be any point on the ellipse.
The equation of tangent at `P(theta)` is
`(xcos theta)/(4)+(y sin theta)(b()=1 " "(1)`
Tangent meets the major axis at `T(a sec theta, 0)`.
Applying since rule in `DeltaPST`, we get
`(PS)/(sin (pi-alpha))=(ST)/(sin beta)`
`or (1-e cos theta)/(sin alpha)=(a(sectheta-e))/(sin beta)`
or `cos theta=(sin alpha)/(sin beta)`

The slop of tangent is `-(b)/(a) cot theta=tan alpha`
`or (b^(2))/(a^(2))=tan^(2)alpha(sec^(2)theta-1)`
`=tan^(2)alpha((sin^(2)beta-sinalpha)/(cos^(2)alpha))`
`=(sin^(2)beta-sin^(2)alpha)/(cos^(2)alpha)`
`or e=sqrt(1-(sin^(2)beta-sin^(2)alpha)/(cos^(2)alpha))`
`=sqrt((1-sin^(2)beta)/(cos^(2)alpha))=(cosbeta)/(cosalpha)`
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