Question

The Cartesian equation of the line passing through the points $(1,-1,2)$ and $(7, 0, 5)$ is

• A. $\frac{x-1}{4}=\frac{y+1}{1}=\frac{z-2}{2}$
• B. $\frac{x-7}{1}=\frac{y}{-1}=\frac{z-5}{2}$
• C. $\frac{x-1}{7}=\frac{y+1}{1}=\frac{z-2}{5}$
• D. $\frac{x-1}{6}=\frac{y+1}{1}=\frac{z-2}{3}$
• E. $\frac{x-7}{6}=\frac{y}{-1}=\frac{z-5}{3}$

Hint:

Geometry Tip: You can immediately eliminate options A and C just by looking at the denominator for $x$, which must be $7-1=6$.

Answer 1

The Correct Option is D

Concept:
The Cartesian equation of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is given by the formula $\frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1}$. The denominators represent the direction ratios of the line.

Step 1: Identify the coordinates of the two points.
Point 1: $(x_1, y_1, z_1) = (1, -1, 2)$ Point 2: $(x_2, y_2, z_2) = (7, 0, 5)$

Step 2: Calculate the direction ratios.
Subtract the coordinates of the first point from the second point: $a = x_2 - x_1 = 7 - 1 = 6$ $b = y_2 - y_1 = 0 - (-1) = 1$ $c = z_2 - z_1 = 5 - 2 = 3$ The direction ratios are $(6, 1, 3)$.

Step 3: Set up the equation using the first point.
Using $(1, -1, 2)$ as the base point $(x_1, y_1, z_1)$ in the Cartesian formula: $$\frac{x - 1}{a} = \frac{y - (-1)}{b} = \frac{z - 2}{c}$$

Step 4: Substitute the direction ratios into the denominators.
Plug in $a=6, b=1, c=3$: $$\frac{x - 1}{6} = \frac{y + 1}{1} = \frac{z - 2}{3}$$

Step 5: Match with the given options.
This expression exactly matches option (D). Note that using the second point $(7,0,5)$ would give $\frac{x-7}{6}=\frac{y}{1}=\frac{z-5}{3}$, but this specific form is not present in the options. Hence the correct answer is (D) $\frac{x-1{6}=\frac{y+1}{1}=\frac{z-2}{3}$}.