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q.1 find the locous of point of intersection of the tangents to the ellipse x sq / a sq + y sq / b sq =1 if the difference of the ecentric angles of their points of contact is 2alpha.

9 years ago

Answers : (1)

vikas askiitian expert
509 Points

let P , Q are two points on ellipse having accentric angle p,q ...


P = [ +(-)acosp , +(-) bsinp ]


Q = [ +(-)acosq  , +(-) bsinq ]


let P,Q : { (acosp , bsinp) , (-acosq,bsinq) }

let point of intersection of tangents is (x1,y1) then chord of contact is given by

xx1/a2 + yy1/b2 = 1                ............................1


slope of this chord is m = -y1a2/x1b2            ............2


slope of this chord can be calculated by poitns (P,Q) , so

m = b[(sinp-sinq)/a(cosp+cosq)]

    =(b/a) [2cos(p+q/2)sin(p-q/2) /2cos(p+q)/2cos(p-q/2) ]

    =(b/a)[ tan(p-q/2) ]

now , p-q = 2alfa            (given) so

m = (b/a)tan(alfa)         ...................3


equating eq2 & 3 we get


y1 =- (b/a)3x1

replace (x1,y1) by x,y we get

y = -(b/a)3 x

this is equation of straight line

9 years ago
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