Ask Uncle Colin: An Implicit problem

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 $2x+2y\frac{dy}{dx}-y-x\frac{dy}{dx}=0$. You could simplify that, but there’s really no need: you know that $\frac{dy}{dx}=\frac{1}{3}$ 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 $(2,10)$ and $(-2,-10)$, 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