Perpendiculars are drawn, respectively, from the points `Pa n dQ` to the chords of contact of the points `Qa n dP` with respect to a circle. Prove that the ratio of the lengths of perpendiculars is equal to the ratio of the distances of the points `Pa n dQ` from the center of the circles.

If the chord of contact of tangents from a point P to a given circle passes through Q , then the circle on P Q as diameter.

The chord of contact of tangents from a point P to a circle passes through Qdot If l_1a n dl_2 are the length of the tangents from Pa n dQ to the circle, then P Q is equal to

If two circles intersect at two points, then prove that their centres lie on the perpendicular bisector of the common chord.

A straight line moves so that the product of the length of the perpendiculars on it from two fixed points is constant. Prove that the locus of the feet of the perpendiculars from each of these points upon the straight line is a unique circle.

Let P be any point on any directrix of an ellipse. Then the chords of contact of point P with respect to the ellipse and its auxiliary circle intersect at (a)some point on the major axis depending upon the position of point Pdot (b)the midpoint of the line segment joining the center to the corresponding focus (c)the corresponding focus (d)none of these

Two parallel tangents to a given circle are cut by a third tangent at the point Ra n dQ . Show that the lines from Ra n dQ to the center of the circle are mutually perpendicular.

Find the length of the chord of contact with respect to the point on the director circle of circle x^2+y^2+2a x-2b y+a^2-b^2=0 .

The lengths of the two tangents drawn from ………….. point to a circle are equal.