ParabolaHardBloom L3
Question
The equation of the other normal to the parabola $y^{2} = 4ax$ which passes through the intersection of those at $(4a, - 4a)$ and $(9a, - 6a)$ is :
Options
A.$5x - y + 115a = 0$
B.$5x + y - 135a = 0$
C.$5x - y - 115a = 0$
D.$5x + y + 115 = 0$
Solution
Sol. [Key Concept: Normals at $t_{1},t_{2}$ meet at a point through which a third normal passes with parameter $t_{3} = - (t_{1} + t_{2})$; write its equation.]
$${t_{1} = - 2,t_{2} = - 3 }{t_{1} + t_{2} + t_{3} = 0 }{\Rightarrow t_{3} = 5 }$$
Equation of normal at $\left( at_{3}^{2},2at_{3} \right)$
$${y = - t_{3}x + 2at_{3} + at_{3}^{3} }{y = - 5x + 10a + 125a }{5x + y - 135a = 0 }$$
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