Question

If three point (h, 0), (a, b) and (0, k) lie on a line, show that $\frac ah+ \frac bk=1$.

Answer 1

If the points A (h, 0), B (a, b), and C (0, k) lie on a line, then

Slope of AB = Slope of BC

$\frac {b-0}{a-h}=\frac {k-b}{0-a}$

⇒ $\frac {b}{a-h} = \frac {k-b}{-a}$

⇒ $-ab=(k-b)(a-h)$

⇒ $-ab=ka-kh-ab+bh$

⇒ $ka+bh=kh$

On dividing both sides by kh, we obtain

$\frac {ka}{kh}+\frac {bh}{kh}=\frac {kh}{kh}$

⇒ $\frac ah+\frac bk=1$

Hence, $\frac ah+\frac bk=1$