Review question

# Can we find the locus of a midpoint created by a parabola? Add to your resource collection Remove from your resource collection Add notes to this resource View your notes for this resource

Ref: R8399

## Suggestion

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

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

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.

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?