Question

If two tangents drawn from the point $P(h,k)$ 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, $a=2\&m_2=3m_1$
$\Rightarrow 4m_1=\frac{k}{h}\&3{\left(m_1\right)}^2=\frac{2}{h}$
$\Rightarrow 3{\left(\frac{k}{4h}\right)}^2=\frac{2}{h}\Rightarrow 3y^2=32x$