Question
If three points $$\left( {h,\,0} \right),\,\left( {a,\,b} \right)$$ and $$\left( {0,\,k} \right)$$ lies on a line, then the value of $$\frac{a}{h} + \frac{b}{k}$$ is :
A.
0
B.
1
C.
2
D.
3
Answer :
1
Solution :
The given points are $$A\left( {h,\,0} \right),\,B\left( {a,\,b} \right),\,C\left( {0,\,k} \right),$$ they lie on the same plane.
$$\therefore $$ Slope of $$AB =$$ Slope of $$BC$$
$$\therefore $$ Slope of $$AB = \frac{{b - 0}}{{a - h}} = \frac{b}{{a - h}}\,;$$
Slope of $$BC = \frac{{k - b}}{{0 - a}} = \frac{{k - b}}{{ - a}}$$
$$\eqalign{
& \therefore \,\frac{b}{{a - h}} = \frac{{k - b}}{{ - a}}\,{\text{ by cross multiplication}} \cr
& {\text{or }} - ab = \left( {a - h} \right)\left( {k - b} \right) \cr
& {\text{or }} - ab = ak - ab - hk + hb \cr
& {\text{or }}0 = ak - hk + hb \cr
& {\text{or }}ak + hb = hk \cr
& {\text{Dividing by }}hk \Rightarrow \frac{{ak}}{{hk}} + \frac{{hb}}{{hk}} = 1{\text{ or }}\frac{a}{h} + \frac{b}{k} = 1 \cr} $$