Can we find the locus of a midpoint created by a parabola?

Can we find the locus of a midpoint created by a parabola?

Question

Show that the tangent to the curve $y = (x+k)^2$ at the point where $x = 2k$ is $y + 3k^2 = 6kx.$

This tangent meets the $x$-axis at $P$ and the $y$-axis at $Q$. The mid-point of $PQ$ is $M$.

Find the co-ordinates of $M$ in terms of $k$ and hence deduce the equation of the locus of $M$ as $k$ varies.

Suggestion

To find the gradient, it might help to expand the bracket.

Given two points in the plane whose coordinates are $(x_1,y_1)$ and $(x_2,y_2)$, can we write down the coordinates of their midpoint?

Solution

We have $y = (x+k)^2 = x^2+2kx+k^2$, and so $\dfrac{dy}{dx}= y' = 2x + 2k$. (If you know the chain rule, you can alternatively use this here.)

This tells us that when $x = 2k, y = 9k^2$, and $y' = 6k$. So the gradient of the tangent is $6k$.

We can write the equation of the tangent as $y = 6kx+c$, and we know that the point $(2k, 9k^2)$ is on the line, so $9k^2 = 12k^2 +c \implies c = -3k^2 .$

Thus the equation is

$y = 6kx - 3k^2$

or, after rearranging,

$$\begin{equation} y + 3k^2 = 6kx, \end{equation}$$

as required.

From ((given-form-for-tangent)), the coordinates of $Q$ are just $(0,-3k^2)$. Similarly, the $x$-coordinate of $P$ is given by $0 = 6kx - 3k^2,$ and so $x = \dfrac{k}{2}$. Thus, the coordinates of $P$ are $\left( \dfrac{k}{2}, 0 \right)$.

The midpoint of the line segment $PQ$ is thus $\left(\dfrac{0+k/2}{2}, \dfrac{-3k^2+0}{2} \right) = \left( \dfrac{k}{4}, -\dfrac{3k^2}{2} \right).$

To find the locus of $M$, we need to eliminate $k$ here. So if $(x,y)$ is on the locus of $M$, $x = \dfrac{k}{4} \implies k = 4x,\quad y = -\dfrac{3k^2}{2} \implies y = -24x^2.$

Now we have the locus of $M$ as $k$ varies; a vertex-up parabola, with its maximum at the origin. You can observe the locus in the following applet by varying $k$.

[interactive embed (geogebra-iframe): //www.geogebratube.org/material/iframe/id/dDg5zVQh/width/513/height/620/border/888888/rc/false/ai/false/sdz/false/smb/false/stb/false/stbh/false/ld/false/sri/true/asb/false/at/html5] — not part of the text import; see the source page.

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