Prove that the ellipse `x^2/a^2 + y^2/b^2 = 1` and the circle `x^2 + y^2 = ab` intersect at an angle `tan^(-1) (|a-b|/sqrt(ab))`.

A circle of radius r is concentric with the ellipse x^2/a^2 + y^2/b^2 = 1 . Prove that the common tangent is inclined to the major axis at an angle tan^(-1) sqrt((r^2-b^2)/(a^2 - r^2) .

Tangents are drawn from a point on the ellipse x^2/a^2 + y^2/b^2 = 1 on the circle x^2 + y^2 = r^2 . Prove that the chord of contact are tangents of the ellipse a^2 x^2 + b^2 y^2 = r^4 .

Tangents are drawn to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1,(a > b), and the circle x^2+y^2=a^2 at the points where a common ordinate cuts them (on the same side of the x-axis). Then the greatest acute angle between these tangents is given by (A) tan^(-1)((a-b)/(2sqrt(a b))) (B) tan^(-1)((a+b)/(2sqrt(a b))) (C) tan^(-1)((2a b)/(sqrt(a-b))) (D) tan^(-1)((2a b)/(sqrt(a+b)))

The locus of the poles of the tangents to the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 w.r.t. the circle x^2 + y^2 = a^2 is: (a) parabola (b) ellipse (c) hyperbola (d) circle

Prove that the curve sin x=cos y does not intersect the circle x^2+y^2=1 at real points.

Prove that area common to ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and its auxiliary circle x^2+y^2=a^2 is equal to the area of another ellipse of semi-axis a and a-b.

AB and CD are two equal and parallel chords of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2)) =1 . Tangents to the ellipse at A and B intersect at P and tangents at C and D at Q. The line PQ

The slopes of the common tangents of the ellipse (x^2)/4+(y^2)/1=1 and the circle x^2+y^2=3 are +-1 (b) +-sqrt(2) (c) +-sqrt(3) (d) none of these

The area of the ellipse (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 is

If the chords through point theta and phi on the ellipse x^2/a^2 + y^2/b^2 = 1 . Intersect the major axes at (c,0).Then prove that tan theta/2 tan phi/2 = (c-a)/(c+a) .