Question

If the three points $(3q,0),(0,3p)$ and $(1,1)$ are collinear, then which one is true?

• A. $\frac{1}{p}+\frac{1}{q}=2$
• B. $\frac{1}{p}+\frac{1}{q}=1$
• C. $\frac{1}{p}+\frac{1}{q}=3$
• D. $\frac{1}{p}+\frac{3}{q}=1$

Answer 1

The correct option is C $\frac{1}{p}+\frac{1}{q}=3$

$A(3q,0),B(0,3p),C(1,1)$ are collinear.
$\therefore$ Slope of $AC=\text{slope of BC}$
$\Rightarrow \frac{1-0}{1-3q}=\frac{1-3p}{1-0}\Rightarrow \frac{1}{1-3q}=\frac{1-3p}{1}$
$\Rightarrow 1=(1-3p)(1-3q)\Rightarrow 1=1-3q-3p+9pq$
$\Rightarrow 3p+3q=9pq\Rightarrow \frac{1}{p}+\frac{1}{q}=3$