A point moves such that the sum of the square of its distances from two fixed straight lines intersecting at angle 2α is a constant. The locus of point is an ellipse of eccentricity

Let us choose the point of intersection of the given lines as the origin and their angular bisector as the x-axis. Then equation of the two lines will be y = mx and y = -mx, where m = tanα.Let P(h, k)be the point whose locus is to be found. Then according to the given condition,PA2 + PB2 = Constant⇒(k−mh)21+m2+(k+mh)21+m2=c (c is a constant) ⇒2k2+m2h2=c1+m2Therefore, the locus of point P isx2a2+y2b2=1,   where a2=c1+m22m2 and b2=c1+m22.  If α<π/4, then m<1, and a2>b2. ∴Eccentricity =1−b2a2=1−m2=cos⁡2αcos⁡α  If α>π/4, then m>1, and a2