Let and for be the foci of the ellipse

MATHEMATICS · Single correct · CBSE · Mathematics · Conic Sections · Conic Sections

Problem

Let

F_1(x_1, 0)

and

F_2(x_2, 0)

for

x_1 < 0

and

x_2 > 0

be the foci of the ellipse

\frac{x^2}{9} + \frac{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. Find the orthocentre of the triangle

F_1MN

. (1)

\left(-\frac{9}{10}, 0\right)

(2)

\left(\frac{2}{3}, 0\right)

(3)

\left(\frac{9}{10}, 0\right)

(4)

\left(\frac{2}{3}, \sqrt{6}\right)