Question

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

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

Answer 1

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

For $y^2=4x$,
here , on comparing we get
$\Rightarrow a=1$
let slope of tangent$=m$
then equation of tangent
$y=mx+\frac{a}{m}\Rightarrow y=mx+\frac{1}{m}$
$\because$ Tangent passes through $(\alpha ,\beta )$
$\beta =m\alpha +\frac{1}{m}$
$\alpha m^2-\beta m+1=0$

Let the slope of one tangent be $p$,
Therefore, slope of other tangent is $2p$
So sum of roots
$\Rightarrow p+2p=\frac{\beta }{\alpha }\Rightarrow 3p=\frac{\beta }{\alpha }\Rightarrow p=\frac{\beta }{3\alpha }\cdots \left(i\right)$
product of roots
$\Rightarrow 2p\times p=\frac{1}{\alpha }\Rightarrow 2p^2=\frac{1}{\alpha }$
Using equation $\left(1\right)$, we get
$2\left(\frac{{\beta }^2}{9{\alpha }^2}\right)=\frac{1}{\alpha }\Rightarrow {\beta }^2=\frac{9\alpha }{2}\Rightarrow \alpha =\frac{2{\beta }^2}{9}$