Question

If two tangents drawn from the point \[(\alpha,\beta)\] to the parabola \[y^{2}=4x\] is such that the slope of one tangent is double of the other, then______.

Options

• \[\beta=\frac{2}{9}\alpha^{2}\]

• \[\alpha=\frac{2}{9}\beta^{2}\]

• \[2\alpha=9\beta^{2}\]

• None of these

Answer 1

If two tangents drawn from the point \[(\alpha,\beta)\] to the parabola \[y^{2}=4x\] is such that the slope of one tangent is double of the other, then \[\alpha=\frac{2}{9}\beta^{2}\].

Explanation:

Any tangent to the parabola \[y^{2}=4x\] is \[y=mx+\frac{1}{m}.\] It passes through \[(\alpha,\beta)\], if 

\[\beta=m\alpha+\frac{1}{m}\]

\[\Rightarrow\quad\alpha m^2-\beta m+1=0\]

It will have roots \[m_{1}\] and \[2m_{1}\], if

\[m_1+2m_1=\frac{\beta}{\alpha}\mathrm{and}m_1\cdot2m_1=\frac{1}{\alpha}\]

\[\Rightarrow\quad2\cdot\left(\frac{\beta}{3\alpha}\right)^2=\frac{1}{\alpha}\Rightarrow\frac{2\beta^2}{9\alpha^2}=\frac{1}{\alpha}\]

\[\therefore\quad\alpha=\frac{2}{9}\beta^{2}\]