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

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 .

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

If the circle x^(2)+y^(2)+2ax=0, a in R touches the parabola y^(2)=4x , them

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 tangent to the parabola y^2 + 4bx = 0 meets the parabola y^2 = 4ax in P and Q. The locus of the middle points of PQ is:

The line 4x+6y+9 =0 touches the parabola y^(2)=4ax at the point

The point of contact of 2x-y+2 =0 to the parabola y^(2)=4ax is

The locus of the mid-point of the chords of the hyperbola x^(2)-y^(2)=4 , that touches the parabola y^(2)=8x is

If x-2y-a=0 is a chord of the parabola y^(2)=4ax , then its langth, is

Tangents are drawn from the points on a tangent of the hyperbola x^2-y^2=a^2 to the parabola y^2=4a xdot If all the chords of contact pass through a fixed point Q , prove that the locus of the point Q for different tangents on the hyperbola is an ellipse.