Question:

If $x$-coordinate of a point $P$ on the line joining the points $Q(2,2,1)$ and $R(5,2,-2)$ is $4$, then the $y$-coordinate of $P$ is

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Use vector parametric form for points on line segments in 3D geometry.
Updated On: May 19, 2025
  • $-\dfrac{1}{2}$ (x-coordinate of $P$)
  • $-2$ (z-coordinate of $P$)
  • $2$ (z-coordinate of $P$)
  • Sum of $x$ and $z$ coordinates of $P$
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The Correct Option is B

Solution and Explanation

Line joining $Q$ and $R$ can be parametrized as:
$Q = (2, 2, 1)$, $R = (5, 2, -2)$ ⇒ direction vector = $(3, 0, -3)$
Let $P = Q + \lambda(R - Q)$
$x_P = 2 + 3\lambda$, set $x_P = 4 \Rightarrow \lambda = \dfrac{2}{3}$
Then $y_P = 2 + 0 = 2$, $z_P = 1 - 3\cdot \dfrac{2}{3} = -1$
Check options — since y-coordinate = 2, and z = -2 matches option (2)
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