Two tangents are drawn from the point (−2,−1) to the parab

Question -

Two tangents are drawn from the point $(- 2, - 1)$ to the parabola $y^{2} = 4 x$ . If $α$ is the angle between those tangents then tan $α$ =

Options:

A .

3

B .

\frac{1}{3}

C . 2

D .

\frac{1}{2}

Answer: Option A
:
A
The equation to the pair of tangents is

[- y - 2 (x - 2)]^{2} = (1 + 8) (y^{2} - 4 x) \Rightarrow (- 2 x - y + 4)^{2} = - (y^{2} - 4 x)

\Rightarrow 4 x^{2} + y^{2} + 16 - 16 x - 8 y + 4 x y = 9 y^{2} - 36 x \Rightarrow 4 x^{2} + 4 x y - 8 y^{2} + 20 x - 8 y + 16 = 0

∴ t a n α = \frac{2 \sqrt{4 + 32}}{| 4 - 8 |} = \frac{2 \times 6}{4} = 3

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