# Let L1 be a tangent to the parabola

Question:

Let $\mathrm{L}_{1}$ be a tangent to the parabola $y^{2}=4(x+1)$ and $\mathrm{L}_{2}$ be a tangent to the parabola $y^{2}=8(x+2)$ such that $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ intersect at right angles. Then $\mathrm{L}_{1}$ and $\mathrm{L}_{2}$ meet on the straight line:

1. (1) $x+3=0$

2. (2) $2 x+1=0$

3. (3) $x+2=0$

4. (4) $x+2 y=0$

Correct Option: 1

Solution:

$L_{1}: y=m_{1}(x+1)+\frac{1}{m_{1}} \quad$ [Tangent to $\left.y^{2}=4(x+1)\right]$

$L_{2}: y=m_{2}(x+2)+\frac{2}{m_{2}} \quad$ [Tangent to $y^{2}=8(x+2)$ ]

$m_{1}^{2}(x+1)-y m_{1}+1=0$

$m_{2}^{2}(x+2)-y m_{2}+2=0$

$\because m_{2}=-\frac{1}{m_{1}} \quad\left(\because L_{1} \perp L_{2}\right)$

[From(ii)]

$\Rightarrow 2 m_{1}^{2}+y m_{1}+(x+2)=0$ (iii)

From (i) and (iii),

$\frac{x+1}{2}=\frac{-y}{y}=\frac{1}{x+2} \Rightarrow x+3=0$