Question

If two tangents drawn from the point $(\alpha ,\beta )$ to the parabola $y^2=4x$ be such that the slope of one tangent is double of the other then

• A. $\beta =\frac{2}{9}{\alpha }^2$
• B. $\alpha =\frac{2}{9}{\beta }^2$
• C. $2\alpha =9{\beta }^2$
• D. $\alpha =2{\beta }^2$

Answer 1

The correct option is B $\alpha =\frac{2}{9}{\beta }^2$

Equation of tangent with slope 'm' to parabola $y^2={\tan }^2$

$y=mx+\frac{a}{m}$

Here, $a=1$

$\Rightarrow my=m^{2}x+a$

$\Rightarrow m^{2}x-my+a=0$

From $(\alpha ,\beta )$

$m^2\alpha -m\beta +a=0$

$\Rightarrow m_1{m}_2=\frac{a}{\alpha }=\frac{1}{\alpha }$

Given $m_2=2m_1$

$\Rightarrow 2m_1^2=\frac{1}{\alpha }$

& $m_1+m_2=\frac{\beta }{\alpha }$

$\Rightarrow 3m_1=\frac{\beta }{\alpha }$

$\Rightarrow m_1=\frac{\beta }{3\alpha }$

$\Rightarrow 2{\left(\frac{\beta }{3\alpha }\right)}^2=\frac{1}{\alpha }$

$\Rightarrow 2\alpha {\beta }^2=9{\alpha }^2$

$\Rightarrow 2{\beta }^2=9\alpha$

$\Rightarrow \left(B\right)$.