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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.
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
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