Question

The equation of a line passing through (-1,6,5) and (-2,4,3), is

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

Hint:

Math Tip: Direction ratios are proportional. The vector $(-1, -2, -2)$ represents the same line as $(1, 2, 2)$. If your standard derivation doesn't match the options, always try multiplying the numerators and/or denominators by $-1$ to check for algebraically equivalent forms!

Answer 1

Answer: (E)

Concept:
Three-Dimensional Geometry - Equation of a Line through Two Points.
The symmetric form of a line passing through two points $(x_1, y_1, z_1)$ and $(x_2, y_2, z_2)$ is: $$ \frac{x - x_1}{x_2 - x_1} = \frac{y - y_1}{y_2 - y_1} = \frac{z - z_1}{z_2 - z_1} $$
Step 1: Identify the given points.
Let point $A(x_1, y_1, z_1)$ be $(-1, 6, 5)$.
Let point $B(x_2, y_2, z_2)$ be $(-2, 4, 3)$.
Step 2: Calculate the direction ratios.
The direction ratios $(a, b, c)$ are given by the differences in coordinates:

• $a = x_2 - x_1 = -2 - (-1) = -1$
• $b = y_2 - y_1 = 4 - 6 = -2$
• $c = z_2 - z_1 = 3 - 5 = -2$

The direction ratios are $(-1, -2, -2)$.
Step 3: Set up the standard equation of the line.
Using point $A(-1, 6, 5)$ and the calculated direction ratios: $$ \frac{x - (-1)}{-1} = \frac{y - 6}{-2} = \frac{z - 5}{-2} $$ $$ \frac{x + 1}{-1} = \frac{y - 6}{-2} = \frac{z - 5}{-2} $$
Step 4: Manipulate the equation to match the given options.
Observe the structures of the options. They feature negative variables in the numerators (e.g., $-x$, $-y$, $-z$).
Multiply the numerator of each fraction by $-1$ to force this form, but do NOT multiply the denominators (as the options leave the denominators intact):

• First term: $\frac{-(x+1)}{-1} \implies \frac{-x-1}{-1}$
• Second term: $\frac{-(y-6)}{-2} \implies \frac{6-y}{-2}$
• Third term: $\frac{-(z-5)}{-2} \implies \frac{-z+5}{-2}$

Step 5: Construct the final matched equation.
Equating the manipulated fractions yields: $$ \frac{-x-1}{-1} = \frac{6-y}{-2} = \frac{-z+5}{-2} $$ This perfectly matches Option E (and Option B, as they are printed identically).