If `A O Ba n dC O D` are two straight lines which bisect one another at right angles, show that the locus of a points `P` which moves so that `P A*P B=P C*P D` is a hyperbola. Find its eccentricity.

If two tangents drawn from a point P to the parabola y^2 = 4x are at right angles, then the locus of P is

If normal at P to a hyperbola of eccentricity e intersects its transverse and conjugate axes at L and M, respectively, then prove that the locus of midpoint of LM is a hyperbola. Find the eccentricity of this hyperbola

A straight line segment AB of length 'a' moves with its ends on the axes. Then the locus of the point P which divides the line in the ratio 1:2 is

The coordinates of a moving point P are (2t^(2) + 4, 4t + 6) , show that the locus of P is a parabola.

Ankita drew two circles which intersect each other at the points P and Q. Through the point P two straight lines are draw so that they intersect one of the circle at the points. A and B and the ohte circle at the points C and D respectively. Prove that angle AQC= angle BQD .

The coordinates of the point A and B are (a,0) and (-a ,0), respectively. If a point P moves so that P A^2-P B^2=2k^2, when k is constant, then find the equation to the locus of the point Pdot

Which one of the following contains P-O-P bond ?

Let the foci of the ellipse (x^(2))/(9) + y^(2) = 1 subtend a right angle at a point P . Then the locus of P is _

Two chords AC and BD of a circle intersect each other at the point O. If two tangents drawn at the points A and B intersect each other at the point P and two tangents drawn at the points C and D intersect at the point Q, prove that angle P+ angle Q =2 angle BOC. [GP-X]

If P N is the perpendicular from a point on a rectangular hyperbola x y=c^2 to its asymptotes, then find the locus of the midpoint of P N