Question
If two tangents drawn from the point $$\left( {\alpha ,\,\beta } \right)$$ 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.
none of these
Answer :
$$\alpha = \frac{2}{9}{\beta ^2}$$
Solution :
Any tangent to the parabola $${y^2} = 4x$$ is $$y = mx + \frac{1}{m}.$$ It passes through $$\left( {\alpha ,\,\beta } \right)$$ if $$\beta = m\alpha + \frac{1}{m}{\text{ or }}\alpha {m^2} - \beta m + 1 = 0$$
It will have roots $${m_1},\,2{m_1}$$ if $${m_1} + 2{m_1} = \frac{\beta }{\alpha },\,\,{m_1}.2{m_1} = \frac{1}{\alpha }\,\,\,\,\,\, \Rightarrow 2.{\left( {\frac{\beta }{{3\alpha }}} \right)^2} = \frac{1}{\alpha }$$