Question

Relation between the slope $m,$ and the parametric variable $t,$ for the point of contact of tangent to the parabola $y^2=4ax$ is

• A. $m=t$
• B. $m=2t$
• C. $m=\frac{1}{t}$
• D. $m=\frac{2}{t}$

Answer 1

The correct option is C $m=\frac{1}{t}$

Given:Parametric variable: $t,$ Slope: $m$

Equation of tangent when slope $m$ is given is $y=mx+c$
and we know when a line $y=mx+c,$ touches the parabola $y^2=4ax,$ then $c=\frac{a}{m}$

$\therefore$ the equation of tangent is $y=mx+\frac{a}{m}...\left(i\right)$

Equation of the tangent when parametric variable $t$ is given is $ty=x+a{t}^2$
$\Rightarrow y=\frac{x}{t}+at...\left(ii\right)\left[\text{divide both sides by t}\right]$

Both equation $\left(i\right),\left(ii\right)$ is representing the same line, so there slope must be equal
$\therefore m=\frac{1}{t}$