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Binomial Theorem

Question
CBSEENMA11015475
Wired Faculty App

The locus of the foot of perpendicular drawn from the centre of the ellipse x2+3y2 =6 on any tangent to it is

  • (x2-y2)2 = 6x2+2y2

  • (x2-y2)2 = 6x2 -2y2

  • (x2+y2)2 = 6x2+2y2

  • (x2+y2)2 = 6x2-2y2

Solution

C.

(x2+y2)2 = 6x2+2y2

Equation of ellipse is,
straight x squared space plus 3 straight y squared space equals space 6 space or space straight x squared over 6 space plus straight y squared over 2 space equals space 1 Equaion space of space the space tangent space is space fraction numerator straight x space cos space straight theta over denominator straight a end fraction plus fraction numerator straight y space sin space straight theta over denominator straight b end fraction space equals space 1 Let space left parenthesis straight h comma straight k right parenthesis space be space any space point space of space the space locus therefore space straight h over straight a cos space straight theta space plus straight k over straight b sinθ space equals space 1 space... space left parenthesis straight i right parenthesis Slope space of space the space tangent space line space is space fraction numerator negative straight b over denominator straight a end fraction cot space straight theta.
The slope of the perpendicular drawn from the centre (0,0) to (h,k) is k/h.
Since, both the lines are perpendicular,
therefore space open parentheses straight k over straight h close parentheses space straight x open parentheses negative straight b over straight a space cot space straight theta close parentheses space equals space minus space 1 rightwards double arrow space cos space straight theta space equals space αha comma space sin space straight theta space equals space αkb From space eq space left parenthesis straight i right parenthesis space we space get straight h over straight a left parenthesis αha right parenthesis space plus space straight k over straight b space left parenthesis αkb right parenthesis space equals space 1 rightwards double arrow space straight h squared straight alpha space plus straight k squared straight alpha space equals space 1 rightwards double arrow straight alpha space equals space fraction numerator 1 over denominator straight h squared plus straight k squared end fraction Also comma space sin squared straight theta space plus space Cos squared space straight theta space equals 1 rightwards double arrow space left parenthesis αkb right parenthesis squared space plus space left parenthesis αha right parenthesis squared space equals 1 rightwards double arrow space straight alpha squared straight k squared straight b squared space plus straight alpha squared straight h squared straight a squared space equals 1 rightwards double arrow fraction numerator straight k squared straight b squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction plus fraction numerator straight h squared straight a squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction space equals 1 fraction numerator 2 straight k squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction plus fraction numerator 6 straight h squared over denominator left parenthesis straight h squared plus straight k squared right parenthesis squared end fraction space equals 1 6 straight x squared plus 2 straight y squared space equals space left parenthesis straight x squared plus straight y squared right parenthesis squared  

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