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))`

If lx+my=1 touches the circle x^(2)+y^(2)=a^(2) , prove that the point (l,m) lies on the circle x^(2)+y^(2)=a^(-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

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.

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)) .

Through a fixed point (h,k), secant are drawn to the circle x^(2)+y^(2)=r^(2) . Show that the locus of the midpoints of the secants by the circle is x^(2)+y^(2)=hx+ky .

At what points on the curve x^2 +y^2 -2x-3= 0 , the tangent is parallel to y-axis?

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 .

Find the equations of the tangents to the circle x^(2) + y^(2)=16 drawn from the point (1,4).

From a point on the line y=x+c , c(parameter), tangents are drawn to the hyperbola (x^(2))/(2)-(y^(2))/(1)=1 such that chords of contact pass through a fixed point (x_1, y_1) . Then , (x_1)/(y_1) is equal to