Dear Uncle Colin,

I have to find the points $A$ and $B$ on the curve $x^2 + y^2 - xy =84$ where the gradient of the tangent is $\frac{1}{3}$. I find four possible points, but the mark scheme only lists two. Where have I gone wrong?

I’ve Miscounted Points Like I Can’t Infer Tangents

Hi, IMPLICIT, and thanks for your message!

I think I have an idea of what’s gone wrong. Let me talk through the question.

Differentiating implicitly

If you differentiate implicitly, you find that $2x + 2y \dydx - y - x \dydx = 0$. You could simplify that, but there’s really no need: you know that $\dydx = \frac{1}{3}$ so you find that $2x + \frac{2}{3} y - y - \frac{1}{3}x =0$, or (multiplying by 3 and simplifying), $5x - y = 0$.

Substituting back in

The simplest thing I can see is to let $y= 5x$ in the original equation, so we get $x^2 + 25x^2 - 5x^2 = 84$, which simplifies to $21x^2 = 84$, or $x^2 = 4$. Thus, $x = \pm 2$.

Here’s where I think your problem probably lies: if you put those two $x$ values into the original equation, you get four possible solutions – however, only two of them also satisfy the second equation.

Using the second equation, you find that $A$ and $B$ are $\br{2, 10}$ and $\br{-2,-10}$, in either order.

Here’s a picture of what’s going on:

The red points are the phantom solutions - it’s easy to see that the gradient at both points is greater than $\frac{1}{3}$.

Hope that helps!

- Uncle Colin