InterviewSolution
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InterviewSolution
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A tangenttotheellipsedistancedistancefromthe centreoftheellipsex^(2) +2y^(2) =6 P andQprovethatthetangent at P andQof theellipsex^(2)+ 2y^(2) =6areat rightangles . |
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Answer» Solution :GIVEN , `x^(2)+ 4y^(2)=4 or(x^(2))/(4)+(y^(2))/(1)=1` Equation of anytangentto theellipseon(i) can bewritten as `(x)/(2)costheta + y sintheta =1` EQUATIONOF second ellipse is  `x^(2)+2y^(2)=6` `implies (x^(2))/(6)+(y^(2))/(3)=1` Supposethe tangent at P andQ meetsat (h,K)Equationof thechordof contactof thetangentsthrough A (h,k) is `(hx)/(6)+(ky)/(3)=1` ButEqs . (iv)and (ii)represent the same STRAIGHTLINE , socomparing Eqs. (iv)adn (ii)we get `(h//6)/( cos theta//2)=(k//3)/( sintheta)=(1)/(1)` `implies h= 3 cos theta andk=3 sintheta` therefore , coordinates of A are( 3 cos,`theta,3 sin theta)` Now , thejointequationof thetangents At A isgivenby `T^(2)=SS_(1)`, `i.e., ((hx)/(6)+(ky)/(3)-1)^(2)=((x^(2))/(6)+(y^(2))/(3)-1)((h^(2))/(6)+(h^(2))/(3)-1)` in Eq. (V)Coefficient of `x^(2)=(h^(2))/(36)-(1)/(6)((h^(2))/(6)+(h^(2))/(3)-1))` `=(h^(2))/(36)-(h^(2))/(36)-(k^(2))/(18)+(1)/(6)=(1)/(6)-(k^(2))/(18)` andcoefficient of `y^(2)=(k^(2))/(9)-(1)/(3)((h^(2))/(6)+(k^(2))/(3)-1)` `=(k^(2))/(9)-(h^(2))/(18)-(k^(2))/(9)+(1)/(3)=-(h^(2))/(18+(1)/(3)` Again , coefficient of `x^(2)+` coefficient of `y^(2)` `=-(1)/(18)(h^(2)+k^(2))+(1)/(6)+(1)/(3)` `=-(1)/(18)( 9 cos ^(2)theta+ 9sin ^(2)theta+(1)/(2)` `=-(9)/(18)+(1)/(2)=0` whichshowsthattwo linesrepresent by Eq. (v)are atrightanglesto eachother. |
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