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

F_(1) and F_(2) are the two foci of the ellipse (x^(2))/(9) + (y^(2))/(4) = 1. Let P be a point on the ellipse such that |PF_(1) | = 2|PF_(2)| , where F_(1) and F_(2) are the two foci of the ellipse . The area of triangle PF_(1)F_(2) is :

F_(1) and F_(2) are the two foci of the ellipse (x^(2))/(9) + (y^(2))/(4) = 1. Let P be a point on the ellipse such that |PF_(1) | = 2|PF_(2)| , where F_(1) and F_(2) are the two foci of the ellipse . The area of triangle PF_(1)F_(2) is :

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

Let F_1(x_1,0)" and "F_2(x_2,0) , for x_1 lt 0 " and" x_2 gt 0 , be the foci of the ellipse (x^2)/(9)+(y^2)/(8)=1 . Suppose a parabola having vertex at the origin 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 the area of the triangle MQR to area of the quadrilateral MF_1NF_2 is :

Let P be a variable on the ellipse (x^(2))/(25)+ (y^(2))/(16) =1 with foci at F_(1) and F_(2)

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.

P is any variable point on the ellipse 4x^(2) + 9y^(2) = 36 and F_(1), F_(2) are its foci. Maxium area of trianglePF_(1)F_(2) ( e is eccentricity of ellipse )

If F_1" and "F_2 be the feet of perpendicular from the foci S_1" and "S_2 of an ellipse (x^2)/(5)+(y^2)/(3)=1 on the tangent at any point P on the ellipse then (S_1F_1)*(S_2F_2) is

Let d be the perpendicular distance from the centre of the ellipse x^2/a^2+y^2/b^2=1 to the tangent drawn at a point P on the ellipse. If F_1 & F_2 are the two foci of the ellipse, then show the (PF_1-PF_2)^2=4a^2[1-b^2/d^2] .

A circle whose diameter is major aixs of ellipe (x^(2))/(a^(2))+(y^(2))/(b^(2))=1 (agtbgt0) meets minor axis at point P. Ifthe orthocentre of DeltaPF_(1)F_(2) lies on ellipse where F_(1)and F_(2) are foci of ellipse , then find the eccenricity of the ellipse