Question

If the points $P(4,5,x)$, $Q(3,y,4)$ and $R(5,8,0)$ are collinear, then the value of $x+y$ is

• A. 6
• B. 7
• C. 4
• D. 5

Hint:

Notice the $x$-coordinates are 4, 3, and 5. Rearranging the points in order along the line gives $Q(3)$, $P(4)$, $R(5)$. Since 4 is the exact midpoint of 3 and 5, point $P$ must be the exact geometric midpoint of segment $QR$! Therefore, $5 = \frac{y+8}{2} \implies y=2$ and $x = \frac{4+0}{2} \implies x=2$.

Answer 1

The Correct Option is C

Step 1: Understanding the Question:
We are given three points in 3D space: $P(4,5,x)$, $Q(3,y,4)$, and $R(5,8,0)$. We are told that these three points are collinear (they lie along the same straight line). We need to determine the value of the combined scalar expression $x+y$.

Step 2: Key Formula or Approach:
If three points $P$, $Q$, and $R$ are collinear, the vectors formed between them, such as $\vec{PQ}$ and $\vec{PR}$, must be parallel. This means their corresponding directional components must be directly proportional: $$\frac{Q_x - P_x}{R_x - P_x} = \frac{Q_y - P_y}{R_y - P_y} = \frac{Q_z - P_z}{R_z - P_z}$$

Step 3: Detailed Explanation:
Let's write down the directional coordinate differences for our ratios: 1. Components for vector direction from $P$ to $Q$: $$\Delta_{PQ} = (3 - 4, \, y - 5, \, 4 - x) = (-1, \, y - 5, \, 4 - x)$$ 2. Components for vector direction from $P$ to $R$: $$\Delta_{PR} = (5 - 4, \, 8 - 5, \, 0 - x) = (1, \, 3, \, -x)$$ Since the vectors are parallel, set up the component proportionality ratios: $$\frac{-1}{1} = \frac{y - 5}{3} = \frac{4 - x}{-x}$$ This breaks down into two separate equations we can solve independently: $$\text{Equation 1: } -1 = \frac{y - 5}{3}$$ Multiply by 3: $$-3 = y - 5 \implies y = 5 - 3 = 2$$ $$\text{Equation 2: } -1 = \frac{4 - x}{-x}$$ Multiply by $-x$: $$x = 4 - x \implies 2x = 4 \implies x = 2$$ Now, compute the required sum $x+y$: $$x + y = 2 + 2 = 4$$

Step 4: Final Answer:
The combined value of $x+y$ is equal to 4, which corresponds to option (C).