Suppose that the foci of the ellipse `(x^2)/9+(y^2)/5=1` are `(f_1,0)a n d(f_2,0)` where `f_1>0a n df_2<0.` Let `P_1a n dP_2` be two parabolas with a common vertex at (0, 0) and with foci at `(f_1 .0)a n d` (2f_2 , 0), respectively. Let`T_1` be a tangent to `P_1` which passes through `(2f_2,0)` and `T_2` be a tangents to `P_2` which passes through `(f_1,0)` . If `m_1` is the slope of `T_1` and `m_2` is the slope of `T_2,` then the value of `(1/(m_1^ 2)+m_2^ 2)` is

Suppose that the foci of the ellipse (x^(2))/(9)+(y^(2))/(5)=-1 are (f_(1),0) and (f_(2),0) where f_(1)gt0 and f_(2)lt0 . Let P_(1) and P_(2) be two parabolas with a common vertex at (0, 0) and with foci at (f_(1),0) and (2f_(2),0) respectively. Let T_(1) be a tangent to P_(1) which passes through (2f_(2),0) and T_(2) be a tangent to P_(2) which passes Through (f_(1),0) . If m_(1) is the slope of T_(1) and m_(2) is the slope of T_(2) , then the value of ((1)/(m_(1)^(2))+m_(2)^(2)) is -

Let F_(1)(x_(1),0) and F_(2)(x_(2),0) , for x_(1)lt0 and x_(2)gt0 , be the foci of the ellipse (x^(2))/(9)+(y^(2))/(8)=1 . Suppose a parabola having vertex at the orgin and focus at F_(2) intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. The orthocentre of the triangle F_(1)MN is

Let F_(1)(x_(1),0) and F_(2)(x_(2),0) , for x_(1)lt0 and x_(2)gt0 , be the foci of the ellipse (x^(2))/(9)+(y^(2))/(8)=1 . Suppose a parabola having vertex at the orgin and focus at F_(2) intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. If the tangents to the ellipse at M and N meet at R and the normal to the parabola at M meets the x-axis at Q, then the ratio of area of the triangle MQR to area of the quadrilateral MF_(1)NF_(2) is

If F_1 and F_2 are the feet of the perpendiculars from the foci S_1a n dS_2 of the ellipse (x^2)/(25)+(y^2)/(16)=1 on the tangent at any point P on the ellipse, then prove that S_1F_1+S_2F_2geq8.

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 curves 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

The n t h derivative of the function f(x)=1/(1-x^2)[w h e r ex in (-1,1)] at the point x=0 where n is even is (a) 0 (b) n ! (c) n^n C_2 (d) 2^n C_2

Suppose that f(x) is a function of the form f(x)=(ax^(8)+bx^(6)+cx^(4)+dx^(2)+15x+1)/(x), (x ne 0)." If " f(5)=2 , then the value of f(-5) is _________.

Suppose that F(n + 1) = (2F(n) + 1)/2 for n = 1,2,3,,,, and F(1) = 2. Then F(101) is

Let f(x)=int(x^2dx)/((1+x^2)(1+sqrt(x^2+1)))a n df(0)=0. Then value of f(1) will be

Which of the following pairs of expression are defined for the same set of values of x ? f_1(x)=2(log)_2xa n df_2(x)=(log)_(10)x^2 f_1(x)=(log)x_xx^2a n df_2(x)=2 f_1(x)=(log)_(10)(x-2)+(log)_(10)(x-3)a n df_(2(x))=(log)_(10)(x-2)(x-3)dot