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:
Points P(4,5,x), Q(3,y,4), R(5,8,0) are collinear; find x+y.

Step 2: Key Formula or Approach:
For collinear points, vectors PQ and PR are parallel, so their component ratios are equal.

Step 3: Detailed Explanation:
PQ = (–1, y–5, 4–x); PR = (1, 3, –x). Ratios: –1/1 = (y–5)/3 = (4–x)/(–x). From –1 = (y–5)/3 → y=2. From –1 = (4–x)/(–x) → x=2. Sum = 4.

Step 4: Final Answer:
x+y = 4, matching option (C).