Three normals are drawn from the point (7, 14) to the parabola `x^2-8x-16 y=0` . Find the coordinates of the feet of the normals.

If normal are drawn from a point P(h , k) to the parabola y^2=4a x , then the sum of the intercepts which the normals cut-off from the axis of the parabola is

Prove that the focal distance of the point (x ,y) on the parabola x^2-8x+16 y=0 is |y+5|

Three normals are drawn from the point (c, 0) to the curve y^2 = x . Show that c must be greater than 1/2. One normal is always the axis. Find c for which the other two normals are perpendicular to each other.

Tangent are drawn from the point (-1,2) on the parabola y^2=4x . Find the length that these tangents will intercept on the line x=2.

A point P moves such that sum of the slopes of the normals drawn from it to the hyperbola xy=16 is equal to the sum of ordinates of feet of normals. The locus of P is a curve C

If the normals from any point to the parabola y^2=4x cut the line x=2 at points whose ordinates are in AP, then prove that the slopes of tangents at the co-normal points are in GP.

If three distinct normals can be drawn to the parabola y^2-2y=4x-9 from the point (2a ,b) , then find the range of the value of adot

Normals are drawn at points A, B, and C on the parabola y^2 = 4x which intersect at P. The locus of the point P if the slope of the line joining the feet of two of them is 2, is

Tangent drawn from the point P(4,0) to the circle x^2+y^2=8 touches it at the point A in the first quadrant. Find the coordinates of another point B on the circle such that A B=4 .

Tangents are drawn from any point on the line x+4a=0 to the parabola y^2=4a xdot Then find the angle subtended by the chord of contact at the vertex.