Paragraph: Let F₁ (x₁, 0 ) and F₂ (x₂, 0 ) , for x₁ 0 , be the foci of the ellipse x² 9 + y² 8 =1 . Suppose a parabola having vertex at the origin and focus at F₂ intersects the ellipse at point M in the first quadrant and at point N in the fourth quadrant. Question: 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

Step-by-step solution

Equation of chord of contact MN is x = 3 2 Let R(h,k) ⇒ Equation of chord of contact ⇒ x h 9 + y k 8 = 1 Comparing we get R(6,0) Normal to parabola at M   : y -   6 =   - 6 2 · 1   x - 3 2 Solving it with y = 0 , we get Q   ≡ 7 2 ,   0 ∴ Area of triangle MQR = 1 2 .   6 - 7 2 .   6 = 5 6 4 Area of quadrilateral M F 1 N F 2 = 2 · 1 2 1 - - 1 6 = 2 6 Required ratio =   5   :   8