Question

The angle between the tangents drawn from the origin to the parabola $y^2 = 4a (x - a)$ is

• A. $ 90^{\circ}$
• B. $ 30^{\circ}$
• C. $tan^{-1} \frac{1}{2}$
• D. $ 45^{\circ}$

Answer 1

Answer: (A)

Given parabola is $y^2 =4a (x - a)$
Equation of tangent : $SS_1 = T^2$
$[y^2 - 4a (x - a)] [y_1^ 2 - 4a (x_1 - a)]$
$= \left[yy_{1} - 4a\left(\frac{x+x_{1}}{2}-a\right)\right]^{2}$
At $x_{1} = 0, y_{1} = 0$, we have
$\left(y^{2} - 4ax + 4a^{2}\right) 4a^{2} = \left(-2ax + 4a^{2}\right)^{2}$
$\Rightarrow y^{2 }- 4ax + 4a^{2} = x^{2 }+ 4a^{2} - 4ax \Rightarrow x^{2} - y^{2} = 0$
$\Rightarrow tan\,\theta = \frac{2\sqrt{h^{2}-ab}}{a+b} = \frac{2\sqrt{0-1.\left(-1\right)}}{1+\left(-1\right)} = \infty$
$\Rightarrow \theta = 90^{\circ}$