If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and…

If the line $\frac{x-1}{2}=\frac{y+1}{3}=\frac{z-1}{4}$ and $\frac{x-3}{1}=\frac{y-\mathrm{k}}{2}=\frac{z}{1}$ intersect; then the value of k is
  1. $\frac{3}{2}$
  2. $\frac{9}{2}$
  3. $-\frac{2}{9}$
  4. $-\frac{3}{2}$

Solution

$\begin{aligned} & \therefore \quad\left|\begin{array}{ccc}x_2-x_1 & y_2-y_1 & z_2-z_1 \\ a_1 & b_1 & c_1 \\ a_2 & b_2 & c_2\end{array}\right|=0 \\ & \therefore \quad\left|\begin{array}{ccc}3-1 & k+1 & 0-1 \\ 2 & 3 & 4 \\ 1 & 2 & 1\end{array}\right|=0 \\ & \therefore \quad\left|\begin{array}{ccc}2 & \mathrm{k}+1 & -1 \\ 2 & 3 & 4 \\ 1 & 2 & 1\end{array}\right|=0 \\ & \therefore \quad \mathrm{k}=\frac{9}{2}\end{aligned}$

Asked in: MHT CET 2024 (10 May Shift 2)

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