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))`

Chords of the circle x^(2)+y^(2)=a^(2) toches the hyperbola x^(2)/a^(2)-y^(2)//b^(2)=1 . Prove that locus of their middle point is the curve (x^2 +y^2)^2=a^(2)x^(2)-b^(2)y^(2)

If from any point on the circle x^(2)+y^(2)=a^(2) tangents are drawn to the circle x^(2)+y^(2)=b^(2), (a>b) then the angle between tangents is

From points on the circle x^2+y^2=a^2 tangents are drawn to the hyperbola x^2-y^2=a^2 . Then, the locus of mid-points of the chord of contact of tangents 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

Tangents are drawn from any point on the hyperbola (x^(2))/(9)-(y^(2))/(4)=1 to the circle x^(2)+y^(2)=9 .If the locus of the mid-point of the chord of contact is a a(x^(2)+y^(2))^(2)=bx^(2)-cy^(2) then the value of (a^(2)+b^(2)+c^(2))/(7873)

A tangent at a point on the circle x^(2)+y^(2)=a^(2) intersects a concentric circle C at two points P and Q. The tangents to the circle X at P and Q meet at a point on the circle x^(2)+y^(2)=b^(2). Then the equation of the circle is x^(2)+y^(2)=abx^(2)+y^(2)=(a-b)^(2)x^(2)+y^(2)=(a+b)^(2)x^(2)+y^(2)=a^(2)+b^(2)

Tangents are drawn from a point on the circle x^(2)+y^(2)=11 to the hyperbola x^(2)/(36)-(y^(2))/(25)=1 ,then tangents are at angle:

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)