If P is the length of the perpendicular from a focus upon the tangent at any point P of the $$ellipsex2/a2+y2/b2=1$$ and r r is the distance of P from the focus, then $$2ar$$−$$b2p2=$$

Question

If P is the length of the perpendicular from a focus upon the tangent at any point P of the $$ellipsex2/a2+y2/b2=1$$ and  r r is the distance of P from the focus,  then $$2ar$$−$$b2p2=$$

Infinity Learn · Accepted Answer

The equation of the tangent at $$P(acos$$⁡θbsin⁡θ$$)t$$   to the ellipse  $$x2/a2+y2/b2=1$$  isxacos⁡θ$$+ybsin$$⁡θ$$=1$$ length of the perpendicular from the focus (ae$$, $$0) on the line is $$p=ecos$$⁡θ−$$1cos2$$⁡θ$$a2+sin2$$⁡θ$$b2=ab(ecos$$⁡θ−$$1)b2cos2$$⁡θ$$+a21$$−$$cos2$$⁡θ$$=ab(ecos$$⁡θ−$$1)a2$$−$$a2e2cos2$$⁡θ$$=b1$$−ecos⁡θ$$1+ecos$$⁡θ$$b2p2=1+ecos$$⁡θ$$1$$−ecos⁡θ Now,   $$r2=(ae$$−acos⁡θ$$)2+b2sin2$$⁡θ$$=a2(e$$−$$cos$$⁡θ$$)2+1$$−$$e2sin2$$⁡θ$$=a2e2cos2$$⁡θ−$$2ecos$$⁡θ$$+1=a2(1$$−ecos⁡θ$$)2$$ ⇒          $$r=a(1$$−ecos⁡θ$$)$$  Now $$2ar$$−$$b2p2=21$$−ecos⁡θ−$$1+ecos$$⁡θ$$1$$−ecos⁡θ$$=1$$

Ellipse

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If $P$ is the length of the perpendicular
from a focus upon the tangent at any point P of the ellipse
$x^{2} / a^{2} + y^{2} / b^{2} = 1$ and $r$ r is the distance of P from the focus,
then $\frac{2 a}{r} - \frac{b^{2}}{p^{2}} =$

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Ellipse

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If P is the length of the perpendicular from a focus upon the tangent at any point P of the ellipsex2/a2+y2/b2=1 and r r is the distance of P from the focus, then 2ar−b2p2=

Ready to Test Your Skills?

free study material

If $P$ is the length of the perpendicular
from a focus upon the tangent at any point P of the ellipse
$x^{2} / a^{2} + y^{2} / b^{2} = 1$ and $r$ r is the distance of P from the focus,
then $\frac{2 a}{r} - \frac{b^{2}}{p^{2}} =$