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

• A. \(-\dfrac{1}{2}\times\) (x-coordinate of \(P\))
• B. \(-2\times\) (z-coordinate of \(P\))
• C. \(2\times\) (z-coordinate of \(P\))
• D. Sum of \(x\) and \(z\) coordinates of \(P\)

Hint:

For a line joining two points in 3D geometry, first form the parametric equations and then substitute the given coordinate condition to find the unknown coordinates.

Answer 1

The Correct Option is B

Step 1: Write the line through $Q$ and $R$ parametrically.
With $Q(2,2,1)$ and $R(5,2,-2)$, the direction ratios are $(5-2,\,2-2,\,-2-1) = (3,0,-3)$. So a point on the line is $x = 2+3t$, $y = 2$, $z = 1-3t$.
Step 2: Use the given $x$-coordinate.
We are told the $x$-coordinate of $P$ is $4$, so $2 + 3t = 4$, giving $3t = 2$ and $t = \frac{2}{3}$.
Step 3: Find the full coordinates of $P$.
Then $y = 2$ and $z = 1 - 3\cdot\frac{2}{3} = 1 - 2 = -1$. So $P = (4,\,2,\,-1)$.
Step 4: Identify the $y$-coordinate.
The $y$-coordinate of $P$ is $2$. We now check which option it equals.
Step 5: Test the relation with $z$.
The $z$-coordinate is $-1$, and $-2\times(-1) = 2$, which is exactly the $y$-coordinate.
Step 6: State the conclusion.
Hence the $y$-coordinate of $P$ equals $-2$ times the $z$-coordinate of $P$.
\[ \boxed{-2\times(z\text{-coordinate of }P)} \]