QJEEAdv16Mac016: JEE Advanced 2016 Mathematics - Class 11 JEEAdv16Mac016 Solution: Let and for , be the foci of the ellipse . Suppose a parabola…

F₁(x, 0) and F₂(x₂, 0) are the foci of the ellipse:

${{{x^2}} \over 9} + {{{y^2}} \over 8} = 1$

Therefore, a² = 9 and b² = 8.

${b^2} = {a^2}(1 - {e^2})$

$1 - {e^2} = {8 \over 9} \Rightarrow {e^2} = 1 - {8 \over 9} = {1 \over 9} \Rightarrow e = {1 \over 3}$

The focus is

${F_1}\left( { - 3 \times {1 \over 3},0} \right)$ and ${F_2}\left( {3 \times {1 \over 3},0} \right)$

That is, F₁( $-$ 1, 0) and F₂(1, 0).

The equation of parabola is

${y^2} = 4(O{F_2})x$

${y^2} = 4x(O{F_2} = 1)$

The point of intersection of ellipse and parabola is

${{{x^2}} \over 9} + {{4x} \over 8} = 1 \Rightarrow {{{x^2}} \over 9} + {x \over 2} = 1$

$\Rightarrow 2{x^2} + 9x - 18 = 0$

$\Rightarrow 2{x^2} + 12x - 3x - 18 = 0$

$\Rightarrow 2x(x + 6) - 3(x + 6) = 0$

$\Rightarrow x = {3 \over 2}$ (x $-$ 6 is rejected)

Now, ${y^2}(4){3 \over 2} = 6$

$y = \pm \sqrt 6$

That is, the points M and N are, respectively, $M\left( {{3 \over 2},\sqrt 6 } \right)$ and $N\left( {{3 \over 2}, - \sqrt 6 } \right)$ .

Let the orthocenter be (h, k).

The slope of $OM = {{k - \sqrt 6 } \over {h - (3/2)}}$

The slope of $ON = {{\sqrt 6 } \over { - 1 - (3/2)}} = {{ - 2\sqrt 6 } \over 5}$

Now, $\left( {{{k - \sqrt 6 } \over {h - (3/2)}}} \right)\left( {{{ - 2\sqrt 6 } \over 5}} \right) = - 1$

$2\sqrt 6 k - 12 = 5h - {{15} \over 2}$

$5h - 2\sqrt 6 k = {{15} \over 2} - 12 = {{ - 9} \over 2}$

The slope of $ON = {{k + \sqrt 6 } \over {h - (3/2)}}$

The slope of ${F_1}M = {{\sqrt 6 } \over {1 + (3/2)}} = {{2\sqrt 6 } \over 5}$

${{k + \sqrt 6 } \over {h - (3/2)}} \times {{2\sqrt 6 } \over 5} = - 1$

$2\sqrt 6 k + 12 = - 5h + {{15} \over 2}$

$5h + 2\sqrt 6 k = {{15} \over 2} - 12 = {{ - 9} \over 2}$

$5h + 2\sqrt 6 k = {{ - 9} \over 2}$ ....... (1)

$5h - 2\sqrt 6 k = {{ - 9} \over 2}$ ........ (2)

Solving Eqs. (1) and (2), we get

$10h = - 9 \Rightarrow h = {{ - 9} \over {10}}$ and k = 0

Hence, the orthocentre of the triangle F₁MN is $\left( {{{ - 9} \over {10}},0} \right)$ .