Question

Which one of the following points lies on the straight line $\frac{x-1}{2}=\frac{y+1}{4}=\frac{z-2}{-2}$?

• A. (2,6,-2)
• B. (4, 3, 1)
• C. (3, 4,-1)
• D. (3,3,0)
• E. (6,2,-1)

Hint:

Geometry Tip: Testing multiple choice coordinates by plugging them in is usually much faster than trying to construct parametric points $x=2k+1, y=4k-1, z=-2k+2$ and solving for $k$.

Answer 1

The Correct Option is D

Concept:
For a point to lie on a given line, its $(x, y, z)$ coordinates must satisfy the equation of the line. We can simply substitute the coordinates of each given point into the line equation and check if all three ratios evaluate to the same constant value.

Step 1: State the line equation.
The given equation of the straight line is: $$\frac{x-1}{2}=\frac{y+1}{4}=\frac{z-2}{-2}$$

Step 2: Test the coordinates of Option A and B.
For (A) $(2, 6, -2)$: $\frac{2-1}{2} = \frac{1}{2}$. Next ratio: $\frac{6+1}{4} = \frac{7}{4}$. Since $1/2 \ne 7/4$, point A is not on the line. For (B) $(4, 3, 1)$: $\frac{4-1}{2} = \frac{3}{2}$. Next ratio: $\frac{3+1}{4} = 1$. Since $3/2 \ne 1$, point B is not on the line.

Step 3: Test the coordinates of Option C.
For (C) $(3, 4, -1)$: $\frac{3-1}{2} = \frac{2}{2} = 1$. Next ratio: $\frac{4+1}{4} = \frac{5}{4}$. Since $1 \ne 5/4$, point C is not on the line.

Step 4: Test the coordinates of Option D.
For (D) $(3, 3, 0)$: $x$-ratio: $\frac{3-1}{2} = \frac{2}{2} = 1$ $y$-ratio: $\frac{3+1}{4} = \frac{4}{4} = 1$ $z$-ratio: $\frac{0-2}{-2} = \frac{-2}{-2} = 1$ All three ratios are perfectly equal to 1.

Step 5: Conclude the correct point.
Because substituting $(3, 3, 0)$ into the line equation yields a consistent value across all dimensions ($1 = 1 = 1$), this point lies exactly on the line. Hence the correct answer is (D) (3,3,0).