Question

The equation of the line joining the points $(-3,4,11)$ and $(1,-2,7)$ is

• A. $\frac{x+3}{2}=\frac{y-4}{3}=\frac{z-11}{4}$
• B. $\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}$
• C. $\frac{x+3}{-2}=\frac{y+4}{3}=\frac{z+11}{4}$
• D. $\frac{x+3}{2}=\frac{y+4}{-3}=\frac{z+11}{2}$

Answer 1

The Correct Option is B

To find the equation of the line passing through the points \((-3, 4, 11)\) and \((1, -2, 7)\), we use the vector form of a line in three-dimensional space. The equation for a line passing through the point \((x_1, y_1, z_1)\) with direction ratios \((a, b, c)\) is given by:

\(\frac{x-x_1}{a}=\frac{y-y_1}{b}=\frac{z-z_1}{c}\) 

First, let's determine the direction ratios by subtracting the coordinates of the given points:

• Direction ratio for \(x\): \(1 - (-3) = 4\)
• Direction ratio for \(y\): \(-2 - 4 = -6\)
• Direction ratio for \(z\): \(7 - 11 = -4\)

Thus, the direction ratios are \((4, -6, -4)\). However, these ratios can be simplified by dividing each by 2:

Resulting simplified direction ratios are \((2, -3, -2)\).

Now, substitute these values into the line equation form using the point \((-3, 4, 11)\):

\(\frac{x+3}{2}=\frac{y-4}{-3}=\frac{z-11}{-2}\)

To compare with the given options, we observe that by multiplying each term by -1, we get the correct formation:

\(\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}\)

This matches the given correct option: \(\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}\).

Therefore, the correct option is:

\(\frac{x+3}{-2}=\frac{y-4}{3}=\frac{z-11}{2}\)