Question

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

Answer 1

Since the lines $A(h,0)$,$B(a,b)$ and $C(0,k)$ lie on the same line.

Slope of $AB=$Slope of $BC$

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

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

$\Rightarrow -ab=(a-h)(k-b)$

$\Rightarrow -ab=ak-ab-hk+hb$

$\Rightarrow hk=ak+hb$

Divide both sides by $hk$ we get

$\frac{hk}{hk}=\frac{ak}{hk}+\frac{hb}{hk}$

or $\frac{a}{h}+\frac{b}{k}=1$

Hence proved.