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Prove that in an ellipse, the perpendicu...

Prove that in an ellipse, the perpendicular from a focus upon any tangent and the line joining the centre of the ellipse to the point of contact meet on the corresponding directrix.

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Any point on the ellipse
`(x^(2))/(a^(2))+(y^(2))/(b^(2))=1 ` be P ( `a cos theta , b sin theta)`
the Equation of tangent at point P is given by
`( x cos theta)/( a) +( y sin theta)/(b)=1`
the equation of line perpendicular to tangent is
`( x sin theta)/(b) -( y cos theta)/( a) =lamda`
since , it passes through the focus ( ae,0), then
`(ae sin theta)/(b) -0= lamda `
`implies lamda = (ae sin theta)/(b)`
`therefore ` Equation is `( x sin theta)/( b) - ( y cos theta ) /(a)= (ae sin theta)/( b)`
Equation of line joining centre and point of contact P ( a cos `theta`,b sin `theta `) is
`y= (b)/(a) ( tan theta ) x`
point of intersection Q of Eqs. (i) and (ii) has x coordinate `( a) /(e) ` hence Q lies on the corresponding diectrix `x=(alpha )/( e)`
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