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.

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