If points $( a,\ 0),\ ( 0,\ b)$ and $( x,\ y)$ are collinear, prove that $\frac{x}{a}+\frac{y}{b}=1$.
Given: Points $( a,\ 0),\ ( 0,\ b)$ and $( x,\ y)$ are collinear.
To do: To prove that $\frac{x}{a}+\frac{y}{b}=1$.
Solution:
Given that three points are collinear, $( a,\ 0),\ ( 0,\ b)$ and $( x,\ y)$.
$\because$ The points are collinear, area of triangle formed by these points should be equal to $0$.
$\frac{1}{2}[x_1( y_2-y_3)+x_2( y_3-y_1)+x_3( y_1-y_2)]=0$
$\frac{1}{2}[a[b-y]+0[x-0]+x(0-b)]=0$
$\Rightarrow ab-ay-bx=0$
$\Rightarrow ay+bx=ab$
$\Rightarrow \frac{ay}{ab}+\frac{bx}{ab}=\frac{ab}{ab}$ [On dividing both sides by $ab$]
$\Rightarrow \frac{y}{b}+\frac{x}{a}=1$
Hence proved.
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