Question

Mutually perpendicular tangents $\mathrm{TA}$ and $\mathrm{TB}$ are drawn to $y^2=4ax$, minimum length of $\mathrm{AB}$ is equal to ____.

Answer 1

Determine the minimum length of chord of contact of mutually perpendicular tangents to a parabola.

We know that, the chord of contact of mutually perpendicular tangents to a parabola is always a focal chord.

Let the focus of the parabola $y^2=4ax$ be $\mathrm{S}\left(\mathrm{a},0\right)$.

Let the ends of the latus rectum of the parabola, $y^2=4ax$ be $\mathrm{L}$ and $\mathrm{L}\text{'}$. Therefore, the $x$-coordinates of $\mathrm{L}$ and $\mathrm{L}\text{'}$will be equal to $a$ as the focus of the parabola is $\mathrm{S}\left(\mathrm{a},0\right)$.

Suppose the co-ordinates of $\mathrm{L}$ be $\left(a,b\right)$.

Since $\mathrm{L}$ is a point on the parabola, we have

$\begin{array}{rcl}{b}^2& =& 4a\left(a\right)\\ & \Rightarrow & b^2=4a^2\\ & \Rightarrow & b=\pm 2a\end{array}$

Therefore, the ends of the latus rectum of the parabola are $\mathrm{L}\left(\mathrm{a},2\mathrm{a}\right)$ and $\mathrm{L}\text{'}\left(a,-2a\right)$

Hence, the length of the latus rectum of the parabola is $\mathrm{LL}\text{'}=4a$.

Since the latus rectum is a focal chord having minimum length, therefore minimum length of $\mathrm{AB}$ is equal to the length of the latus rectum $\mathrm{LL}\text{'}$.

Hence, minimum length of $\mathrm{AB}$ is equal to $4a$.