ParabolaHardBloom L3
Question
If the segment intercepted by the parabola $y^{2} = 4ax$ with the line $lx + my + n = 0$ subtends a right angle at vertex, then:
Options
A.$al + n = 0$
B.$4am + n = 0$
C.$4al + n = 0$
D.$4am - n = 0$
Solution
Sol. [Key Concept: Homogenize $y^{2} = 4ax$ with the chord $lx + my + n = 0$; a right angle at the vertex needs (coeff $x^{2}$)+(coeff $y^{2}$)$= 0$, giving $4al + n = 0$.]
$${OA\bot OB }{\Rightarrow \frac{2}{t_{1}}\frac{2}{t_{2}} = - 1 \Rightarrow t_{1}t_{2} = - 4 }$$
Put ( $at^{2},2at$ ) in eqn. of $AB$
$\Rightarrow$ lat $\ ^{2} + 2$ mat $+ n = 0^{- t_{1}}$
$${t_{1}t_{2} = - 4 = \frac{n}{la} }{\Rightarrow n + 4la = 0 }$$
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