Question

If three points A (h, 0), P (a, b) and B (0, k) lie on a line, show that: $\frac{a}{h}+\frac{b}{k}=1$.

Answer 1

The given points are A (h, 0), P (a, b) and B (0, k).
Thus, we have,

$$\begin{gathered} \mathrm{Slope}\mathrm{of}AP=\frac{b-0}{a-h} \\ \mathrm{Slope}\mathrm{of}BP=\frac{b-k}{a-0} \end{gathered}$$

For the given points to be collinear, we must have,

Slope of AP = Slope of BP

$$\begin{gathered} \Rightarrow \frac{b-0}{a-h}=\frac{b-k}{a-0} \\ \Rightarrow \frac{b}{a-h}=\frac{b-k}{a} \\ \Rightarrow ab=ab-ak-bh+hk \\ \Rightarrow ak+bh=hk \\ \Rightarrow \frac{a}{h}+\frac{b}{k}=1\left[\mathrm{On}\mathrm{dividing}\mathrm{both}\mathrm{sides}\mathrm{by}hk\right] \end{gathered}$$