From the points on the circle `x^(2)+y^(2)=a^(2)`, tangents are drawn to the hyperbola `x^(2)-y^(2)=a^(2)`: prove that the locus of the middle-points `(x^(2)-y^(2))^(2)=a^(2)(x^(2)+y^(2))`

From any point on the circle x^(2)+y^(2)=a^(2) tangents are drawn to the circle x^(2)+y^(2)=a^(2) sin^(2) theta . The angle between them is

From any point on the circle x^(2)+y^(2)=a^(2) tangent are drawn to the circle x^(2)+y^(2)=a^(2)sin^(2)theta . The angle between them is

Chords of the circle x^(2)+y^(2)=4 , touch the hyperbola (x^(2))/(4)-(y^(2))/(16)=1 .The locus of their middle-points is the curve (x^(2)+y^(2))^(2)=lambdax^(2)-16y^(2) , then the value of lambda is

A line through the origin meets the circle x^(2)+y^(2)=a^(2) at P and the hyperbola x^(2)-y^(2)=a^(2) at Q. Prove that the locus of the point of intersection of tangent at P to the circle with the tangent at Q to the hyperbola is a straight line.

The locus of poles of tangents to the circle (x-p)^(2)+y^(2)=b^(2) w.r.t. the circle x^(2)+y^(2)=a^(2) is

The tangents from P to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 are mutually perpendicular show that the locus of P is the circle x^(2)+y^(2)=a^(2)-b^(2)

Tangents are drawn from point P on the curve x^(2) - 4y^(2) = 4 to the curve x^(2) + 4y^(2) = 4 touching it in the points Q and R . Prove that the mid -point of QR lies on x^(2)/4 - y^(2) = (x^(2)/4 + y^(2))^(2)

Tangents are drawn from the origin to the curve y = sin x . Prove that their points of contact lie on the curve x^(2) y^(2) = (x^(2) - y^(2))

Tangents are drawn to the hyperbola x^(2)-y^(2)=3 which are parallel to the line 2x+y+8=0 . Then their points of contact is/are :

From a point on the line x-y+2=0 tangents are drawn to the hyperbola (x^(2))/(6)-(y^(2))/(2)=1 such that the chord of contact passes through a fixed point (lambda, mu) . Then, mu-lambda is equal to