Question

If points $(a,0),(0,b)$ and $(x,y)$ are collinear, prove that $\frac{x}{a}+\frac{y}{b}=1$.

Answer 1

If three points $(x_1,y_1),(x_2,y_2)$ and $(x_3,y_3)$ are collinear, then

$\left|\begin{array}{ccc}{x}_1& y_1& 1\\ x_2& y_2& 1\\ x_3& y_3& 1\end{array}\right|=0$

$\therefore \left|\begin{array}{ccc}x& y& 1\\ a& 0& 1\\ 0& b& 1\end{array}\right|=0$

$\Rightarrow -bx-ay+ab=0\Rightarrow bx+ay=ab$

$\Rightarrow \frac{x}{a}+\frac{y}{b}=1$