The center of an ellipse is `C` and `P N` is any ordinate. Point `A ,A '` are the endpoints of the major axis. Then find the value of `(P N^2)/((A N).A^(prime)N)`

If the sum of n terms of an A.P. is 3n^(2) + 5n and its m^(th) term is 164, find the value of m .

Prove that if any tangent to the ellipse is cut by the tangents at the endpoints of the major axis at Ta n dT ' , then the circle whose diameter is TT ' will pass through the foci of the ellipse.

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

A circle concentric with the ellipse (x^2)/(a^2)+(y^2)/(b^2)=1 and passes through the foci F_1a n dF_2 of the ellipse. Two curves intersect at four points. Let P be any point of intersection. If the major axis of the ellipse is 15 and the area of triangle P F_1F_2 is 26, then find the value of 4a^2-4b^2dot

A circle has the same center as an ellipse and passes through the foci F_1a n dF_2 of the ellipse, such that the two cuves intersect at four points. Let P be any one of their point of intersection. If the major axis of the ellipse is 17 and the area of triangle P F_1F_2 is 30, then the distance between the foci is (a) 13 (b) 10 (c) 11 (d) none of these

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.

The probability that at least one of Aa n dB occurs is 0.6. If Aa n dB occur simultaneously with probability 0.3, then find the value of P(A^(prime))+P(B^(prime))dot

P is a point on the hyperbola (x^2)/(a^2)-(y^2)/(b^2)=1,N is the foot of the perpendicular from P on the transverse axis. The tangent to the hyperbola at P meets the transvers axis at Tdot If O is the center of the hyperbola, then find the value of O TxO Ndot

From any point on the line y=x+4, tangents are drawn to the auxiliary circle of the ellipse x^2+4y^2=4 . If P and Q are the points of contact and Aa n dB are the corresponding points of Pa n dQ on he ellipse, respectively, then find the locus of the midpoint of A Bdot