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)`

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

Chords of the hyperbola,x^(2)-y^(2)=a^(2) touch the parabola,y^(2)=4ax. Prove that the locus of their middlepoints is the curve,y^(2)(x-a)=x^(3)

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))

A chord PQ of the hyperbola xy=c^(2) is tangent to the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 Find the locus of the middle point of PQ .

Tangents at right angle are drawn to the ellipse x^(2)/a^(2)+y^(2)/b^(2)=1 . Show that the focus of the middle points of the chord of contact is the curve (x^(2)/a^(2)+y^(2)/b^(2))^(2)=(x^(2)+y^(2))/(a^(2)+b^(2)) .

P is a variable points on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 whose vertex is A(a,0) The locus of the middle points AP is