16. Which diagram represents the set of all points $(x, y)$ satisfying $y^{2}-2 y=x^{2}+2 x$ ?
Solution
E
We have
$$
\begin{aligned}
y^{2}-2 y=x^{2}+2 x & \Leftrightarrow y^{2}-x^{2}-2 y-2 x=0 \\
& \Leftrightarrow(y-x)(y+x)-2(y+x)=0 \\
& \Leftrightarrow(y-x-2)(y+x)=0 \\
& \Leftrightarrow y-x-2=0 \quad \text { or } \quad y+x=0 \\
& \Leftrightarrow y=x+2 \quad \text { or } \quad y=-x .
\end{aligned}
$$

Note: An alternative method is to complete the square on both sides of the equation. See Problem 16.1 below.

We therefore see that the set of points satisfying the equation $y^{2}-2 y=x^{2}+2 x$ is made up of all the points on the line with the equation $y=x+2$ together with all the points on the line with the equation $y=-x$.
The line with the equation $y=x+2$ is the line with slope 1 that goes through the point $(0,2)$.
The line with the equation $y=-x$ is the line with slope -1 that goes through the point $(0,0)$.
The diagram that shows these two lines is that in option E.
For investigation
16.1 Show that adding 1 to both sides of the equation $y^{2}-2 y=x^{2}+2 x$, provides an alternative way to show that $y=x+2$ or $y=-x$.
16.2 What are the coordinates of the point where the lines with equations $y=x+2$ and $y=-x$ meet?
16.3 Draw a diagram to represent the set of all points $(x, y)$ which satisfy the equation $x^{2}-1=y^{2}+2 y$.
16.4 Draw a diagram to represent the set of all points $(x, y)$ which satisfy the equation $x^{2} y^{2}+x y=x^{3}+y^{3}$.