Question

If two tangents drawn from the point $P\left(\right.h,k\left.\right)$ to the parabola $y^{2}=8x$ are such that the slope of one of the tangent is $3$ times the slope of the other, then the locus of point $P$ is

• A. $3y^{2}=16x$
• B. $3y^{2}=8x$
• C. $y^{2}=32x$
• D. $3y^{2}=32x$

Answer 1

Answer: (D)

Equation of tangent in slope form is $y=mx+\frac{a}{m}$
Let the tangent passes through $P\left(h , k\right)$
$\Rightarrow k=mh+\frac{a}{m}\Rightarrow m^{2}h-km+a=0$
$\Rightarrow m_{1}+m_{2}=\frac{k}{h}\&m_{1}m_{2}=\frac{a}{h}$
Where, $a=2\&m_{2}=3m_{1}$
$\Rightarrow 4m_{1}=\frac{k}{h}\&3\left(m_{1}\right)^{2}=\frac{2}{h}$
$\Rightarrow 3\left(\frac{k}{4 h}\right)^{2}=\frac{2}{h}\Rightarrow 3y^{2}=32x$