Question

Let 𝑃 be the midpoint of the line segment joining the points 𝐴(𝑎 + 𝑏, 𝑏) and 𝐵(𝑎 − 𝑏, 𝑎 + 𝑏). Find the slope of the line passing through 𝑃 and 𝑄 (𝑏, −𝑎/2) .

Answer 1

It follows that for the point $P(x,y)$ we have that $x=\frac{a+b+a-b}2=a,\ y=\frac{b+a+b}2=\frac{a}2+b.$ Therefore, the slope of the line passing through $P(a,\frac{a}2+b)$ and $𝑄 (𝑏, −\frac{𝑎}2)$ is equal to

$\frac{\frac{a}2+b-(-\frac{a}2)}{a-b}=\frac{a+b}{a-b}.$