Question

If the points $(a,0),(0,b)$ and $(1,1)$ are collinear, which of the following is true?

• A. $\frac{1}{a}+\frac{1}{b}=2$
• B. $\frac{1}{a}-\frac{1}{b}=1$
• C. $\frac{1}{a}-\frac{1}{b}=2$
• D. $\frac{1}{a}+\frac{1}{b}=1$

Answer 1

The correct option is D $\frac{1}{a}+\frac{1}{b}=1$

The given points $(a,0),(0,b),(1,1)$ are collinear.

This means that $\left|\begin{array}{ccc}1& 1& 1\\ x_1& y_1& {}_1\\ x_2& y_2& z_2\end{array}\right|=0$

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

$1(0-b)-1(a-0)+1(ab-0)=0$

$-b-a+ab=0$

Dividing by $-ab$ on both sides, we get

$\frac{1}{a}+\frac{1}{b}=1$