Paragraph: Let F 1 ( x 1 , 0 ) and F 2 ( x 2 , 0 ) , for x 1 0 , be the foci of the ellipse 9 x 2 + 8 y 2 = 1. Suppose a parabola having vertex at the origin and focus at

Question

Paragraph: 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 9 x 2 ​ + 8 y 2 ​ = 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 ​ MN is

TestHub · Accepted Answer

x 2 9 + y 2 8 = 1 F 1 ( − 1 , 0 ) , F 2 ( 1 , 0 ) ⇒ Parabola is y 2 = 4 x The intersection of ellipse & parabola is M & N x 2 9 + 4 x 8 = 1 ⇒ M ( 3 2 , 6 ) & N ( 3 2 , − 6 ) Equation of altitude through M : y - 6 = 5 2 6 x - 3 2 Equation of altitude through F 1 : y = 0 Solving, we get orthocentre - 9 10 , 0

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