Question

Two tangents are drawn from a ponit $P$ to the parabola $y^2=4x$ such that the slope of one tangent is $5$ times the slope of the other. The locus of $P$ is

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

Answer 1

Answer: (B)

a straight line

We have,

$y^2=4x$

Compare that,

$y^2=4ax$

So,

$a=1$

We know that,

$cm=a,$

$c=\frac{a}{m}$

If $a=1$

Now,

$c=\frac{1}{m}$

We know that, equation of line $y=mx+c$

Put the value of $c$ in equation of line

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

$my=m^{2}x+1$

$m^{2}x-ym+1=0$

Therefore,

$m_1.m_2=\frac{1}{x}$

Also tangents are drawn to parabola at right angles

So,

locus of $P$ is

$x=1$

Hence, this is straight line.