Question

$P$ is a point. Two tangents are drawn from it to the parabola $y^2=4x$ such that the slope of one tangent is three times the slope of the other. The locus of $P$ is

• A. a straight line
• B. a circle
• C. a parabola
• D. an ellipse

Answer 1

The correct option is C a parabola

Let P be $(x_1,y_1)$

$y=mx+\frac{1}{m}$ is $e{q}^n$ of tangent to parabola

It has to pass through $(x_1,y_1)$

$\Rightarrow m^2{x}_1-m{y}_1+1=0$

$m_1=3m_2$

$m_1+m_2=\frac{y_1}{x_1}=4m_2$

$m_1{m}_2=\frac{1}{x_1}=3m{{}_2}^2$

${\left(\frac{y}{4x}\right)}^2=\frac{1}{3x}$

$y^2=\frac{4x}{3}$

Thus the locus of the above point is a parabola.