ParabolaHardBloom L3
Question
Tangents are drawn from the point $( - 1,2)$ on the parabola $y^{2} = 4x$. The length, these tangents will intercept on the line $x = 2$ is :
Options
A.6
B.$6\sqrt{2}$
C.$2\sqrt{6}$
D.none of these
Solution
Sol. [Key Concept: Find the two tangents from the external point (${SS}_{1} = T^{2}$), then evaluate both on the line $x = 2$ and take the length between the meets.]
Equation of pair $PA$ and $PB$ is
$$(y(2) - 2(x - 1))^{2} = \left( y^{2} - 4x \right)(4 + 4)$$
Put $\ x = 2$
$$\begin{matrix} \Rightarrow & \ (y - 1)^{2} = 2\left( y^{2} - 8 \right) \\ \Rightarrow & y^{2} + 2y - 17 = 0 \\ & \left( y_{1} - y_{2} \right)^{2} = ( - 2)^{2} - 4( - 17) = 72 \\ & \ \Rightarrow \left| y_{1} - y_{2} \right| = 6\sqrt{2} \end{matrix}$$
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