The tangent at a point `P` on an ellipse intersects the major axis at `T ,a n dN` is the foot of the perpendicular from `P` to the same axis. Show that the circle drawn on `N T` as diameter intersects the auxiliary circle orthogonally.

N is the foot of the perpendicular form P to the diameter AB of the circle APB. P lies on the circle APB. Prove that PB^2=AB.NB ,

Show that the tangents at the ends of latus rectum of an ellipse intersect on the major axis.

Show that the tangent at any point on an ellipse and the tangent at the corresponding point on its auxiliary circle intersect on the major axis.

The tangent at any point P onthe parabola y^2=4a x intersects the y-axis at Qdot Then tangent to the circumcircle of triangle P Q S(S is the focus) at Q is

P is a point on the hyperbola (x^(2))/(a^(2))-(y^(2))/(b^(2))=1 , and N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transverse axis at T. If O is the centre of the hyperbola, then OT.ON is equal to

P is a point on the straight line 7x-4y-29=0 . If the foot of the perpendicular drawn from P on the line 5x+2y+18=0 be N(-2,-4) , find the coordinates of P.

Two tangents to a circle at A and B intersect at a point P. If angleAPB = 68^@ then find anglePAB .

The two circles with centre X and Y intersect each other at the points A and B, A joined with the mid -point S of XY and the perpendicular on SA through the point A is drawn which intersect the two circles at the point P and Q .Prove that PA = AQ.

The locus of the foot of perpendicular drawn from origin to a variable line passing through fixed points (2,3) is a circle whose diameter is?

The lengths of the major and minor axes of an ellipse are 2 a and 2b and N is the foot of the perpendicular drawn from a point P of the ellipse on the major axis . Show that (PN^(2))/(overline(AN)*overline(A'N))=(b^(2))/(a^(2)) where A and A' are the two vertices of the ellipse.