## My Mathematical Valentines Message

Here’s a little mathematical exercise with a Valentines theme:

Sketch the curve in the x-y plane described by the equation

$\left(x^2 +y^2 + 2ay \right)^2 = 4a^2 \left( x^2 + y^2 \right)$

for

$x<3$.

Geddit?

Answer: the equation is that of a cardioid:

### 5 Responses to “My Mathematical Valentines Message”

1. Oh, dear. Oh, dear. Oh, dear.

(Or have I got it wrong?)

2. Anton Garrett Says:

Rewrite in polar coordinates and it reduces to

r = 2a (1 – sin \theta )

at which point I recognised it from prior knowledge. Nice!

3. Ah. Now Peter has changed the sign in front of the 2ay term. The original version had a different sign – and that didn’t produce anything like a heart for a>0.