Paragraph: Let $F_{1}\left(x_{1}, 0\right)$ and $F_{2}\left(x_{2}, 0\right)$, for $x_{1} 0$, be the foci of…

Paragraph: Let $F_{1}\left(x_{1}, 0\right)$ and $F_{2}\left(x_{2}, 0\right)$, 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.
Question: The orthocentre of the triangle $F_{1} M N$ is
  1. -910, 0
  2. 23, 0
  3. 910, 0
  4. 23, 6

Solution


x 2 9 + y 2 8 =1
F 1 ( −1,0 ), F 2 ( 1,0 )
⇒ Parabola is y 2 =4x
The intersection of ellipse & parabola is M & N
x 2 9 + 4x 8 =1⇒M( 3 2 , 6 ) &  N( 3 2 ,− 6 )

Equation of altitude through M :y- 6=526 x-32
Equation of altitude through F1:y=0
Solving, we get orthocentre -910, 0

Asked in: JEE Advanced 2016 (Paper 2)

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