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 line l x+m y+n=0 touches the parabola y^2=4a x , prove that ln=a m^2

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 straight line touches the rectangular hyperbola 9x^2-9y^2=8 and the parabola y^2= 32x . An equation of the line is

The locus of the middle points of normal chords of the parabola y^2 = 4ax is-

If a chord PQ of the parabola y^2 = 4ax subtends a right angle at the vertex, show that the locus of the point of intersection of the normals at P and Q is y^2 = 16a(x - 6a) .

The circle x^2+y^2+4lamdax=0 which lamda in R touches the parabola y^2=8x . The value of lamda is given by

If a hyperbola be rectangular, and its equation be xy = c^2, prove that the locus of the middle points of chords of constant length 2d is (x^2+y^2)(xy-c^2)=d^2 xy.

The circle x^(2)+y^(2)+2lamdax=0,lamdainR , touches the parabola y^(2)=4x externally. Then,