Q116 / 160 · WBJEE 2010 · Mathematics
If the three points $(3 q, 0),(0,3 p)$ and $(1,1)$ are collinear then which one is true ?
- A$\frac{1}{\mathrm{p}}+\frac{1}{\mathrm{q}}=1$
- B$\frac{1}{\mathrm{p}}+\frac{1}{\mathrm{q}}=1$
- Correct answerC$\frac{1}{\mathrm{p}}+\frac{1}{\mathrm{q}}=3$
- D$\frac{1}{\mathrm{p}}+\frac{3}{\mathrm{q}}=1$
Solution
Hints: $\mathrm{A}(3 \mathrm{q}, 0) \mathrm{B}(0,3 \mathrm{p}) \mathrm{C}(11)$
Slope $=1 \mathrm{AC}=5 \log \mathrm{BC}$
$$
\begin{aligned}
& \frac{1-0}{1-3 q}=\frac{1-3 p}{1-0}=3, \frac{1}{1-3 q}=\frac{1-3 p}{1} \\
& 1=(1-3 p)(1-3 q), 1=1-3 q-3 p+9 p q \\
& \Rightarrow 3 p+3 q=9 p q, \frac{1}{q}+\frac{1}{p}=3
\end{aligned}
$$
Hence, the correct answer is option: 'C'.