Question

If two tangents drawn from the point $(x_1,y_1)$ to the parabola $y^2=4x$ be such that the slope of one tangent is double of the other, then

• A. $2{y_1}^2=9x_1$
• B. $2{x_1}^2=9y_1$
• C. $4{y_1}^2=9x_1$
• D. None of these

Answer 1

The correct option is A $2{y_1}^2=9x_1$

The equation of any tangent to the parabola $y^2=4x$ is

$y=mx+\frac{1}{m}$

If it passes through the point $(x_1,y_1)$, then

$y_1=m{x}_1+\frac{1}{m}\Rightarrow x_1{m}^2-y_{1}m+1=0$

Its roots are given to be $m_1$ and $2m_1$

$\therefore m_1+2m_1=\frac{y_1}{x_1}\Rightarrow 3m_1=\frac{y_1}{x_1}$ and $m_1.2m_1=\frac{1}{x_1}\Rightarrow 2{m_1}^2=\frac{1}{x_1}$

$\therefore 2{\left(\frac{y_1}{3x_1}\right)}^2=\frac{1}{x_1}\Rightarrow 2{y_1}^2=9x_1$