Q231 / 270 · NDA 2022 (Phase 1) · Mathematics
If the points with coordinates $(-5,0),\left(5 p^2, 10 p\right)$ and $\left(5 q^2, 10 q\right)$ are collinear, then what is the value of $p q$ where $p \neq q$ ?
- A-2
- B-1
- Correct answerC1
- D2
Solution
Since $(-5,0),\left(5 p^2, 10 p\right)$ and $\left(5 q^2, 10 q\right)$ are collinear.
$\begin{aligned} & \therefore\left|\begin{array}{ccc}-5 & 0 & 1 \\ 5 p^2 & 10 p & 1 \\ 5 q^2 & 10 q & 1\end{array}\right|=0 \\ & \Rightarrow \quad 5 \times 10\left|\begin{array}{ccc}-1 & 0 & 1 \\ p^2 & p & 1 \\ q^2 & q & 1\end{array}\right|=0\end{aligned}$
Applying $\quad c_1 \rightarrow c_1+c_3$
$\Rightarrow 50\left|\begin{array}{ccc}0 & 0 & 1 \\ p^2+1 & p & 1 \\ q^2+1 & q & 1\end{array}\right|=0$
$\begin{aligned} & \Rightarrow \quad q\left(p^2+1\right)-p\left(q^2+1\right)=0 \\ & \Rightarrow p^2 q+q-p q^2-p=0 \\ & \Rightarrow p^2 q-p q^2+q-p=0 \\ & \Rightarrow p q(p-q)-1(p-q)=0 \\ & \Rightarrow(p-q)(p q-1)=0 \\ & \Rightarrow p q-1=0 \Rightarrow p q=1 \quad(\because p \neq q)\end{aligned}$
Hence, the correct answer is option: 'C'.