Question

Find the locus of the mid-point of the chords of the parabola $y^2 = 4ax$ such that tangent a the extremities of the chords are perpendicular.

Answer 1

The locus of the mid-point of the chords of the parabola $y^2 = 4ax$ such that tangents at the extremities of the chords are perpendicular is $y^2 = 2a(x-a)$.

Answer 2

Let the parabola be $y^2 = 4ax$. Let the extremities of a chord be $P(at_1^2, 2at_1)$ and $Q(at_2^2, 2at_2)$ in parametric form.

The mid-point $M(h, k)$ of the chord $PQ$ is given by: $h = \frac{at_1^2 + at_2^2}{2} = \frac{a}{2}(t_1^2 + t_2^2)$ $k = \frac{2at_1 + 2at_2}{2} = a(t_1 + t_2)$

The slope of the tangent to the parabola $y^2 = 4ax$ at a point $(at^2, 2at)$ is $m = \frac{1}{t}$. The slopes of the tangents at the extremities $P$ and $Q$ are $m_1 = \frac{1}{t_1}$ and $m_2 = \frac{1}{t_2}$ respectively.

The condition given is that the tangents at the extremities of the chords are perpendicular. Therefore, $m_1 m_2 = -1$. $\frac{1}{t_1} \cdot \frac{1}{t_2} = -1$ $t_1 t_2 = -1$.

Now, we need to find the locus of $M(h, k)$ by eliminating $t_1$ and $t_2$ using the expressions for $h$ and $k$ and the condition $t_1 t_2 = -1$.

We know that $t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2t_1 t_2$. Substitute the condition $t_1 t_2 = -1$: $t_1^2 + t_2^2 = (t_1 + t_2)^2 - 2(-1) = (t_1 + t_2)^2 + 2$.

Now, substitute this into the expression for $h$: $h = \frac{a}{2}(t_1^2 + t_2^2) = \frac{a}{2}((t_1 + t_2)^2 + 2)$.

From the expression for $k$, we have $t_1 + t_2 = \frac{k}{a}$. Substitute this into the equation for $h$: $h = \frac{a}{2}\left(\left(\frac{k}{a}\right)^2 + 2\right)$ $h = \frac{a}{2}\left(\frac{k^2}{a^2} + 2\right)$ $h = \frac{a}{2}\left(\frac{k^2 + 2a^2}{a^2}\right)$ $h = \frac{k^2 + 2a^2}{2a}$.

Rearranging this equation to express the locus in terms of $h$ and $k$: $2ah = k^2 + 2a^2$ $k^2 = 2ah - 2a^2$ $k^2 = 2a(h - a)$.

Replacing $(h, k)$ with $(x, y)$ to represent the locus in standard Cartesian coordinates: $y^2 = 2a(x - a)$.

This is the equation of the locus of the mid-point of the chords of the parabola $y^2 = 4ax$ such that the tangents at the extremities of the chords are perpendicular.