Question

Let the line \( L \) pass through the point \( (-3, 5, 2) \) and make equal angles with the positive coordinate axes. If the distance of \( L \) from the point \( (-2, r, 1) \) is \( \sqrt{\frac{14}{3}} \), then the sum of all possible values of \( r \) is :

• A. 16
• B. 12
• C. 10
• D. 6

Hint:

For a line making equal angles with axes, the direction ratios are always \( 1 : 1 : 1 \).

Answer 1

The Correct Option is C

To solve this problem, we need to analyze the properties of the line \( L \), which passes through point \((-3, 5, 2)\) and makes equal angles with the positive coordinate axes. 

The direction ratios of a line that makes equal angles with all coordinate axes are proportional to \((1, 1, 1)\) or \((-1, -1, -1)\). Hence, the direction vector of the line \( L \) is \(\vec{d} = (a, a, a)\). Since making equal angles implies all direction cosines are equal, by normalization, we can assume the direction vector as \((1, 1, 1)\).

The parametric form of a line passing through a point \((x_0, y_0, z_0)\) with direction vector \((a, b, c)\) is given by:

1. \(x = x_0 + ta, \, y = y_0 + tb, \, z = z_0 + tc.\)

For this line, the parametric equations become:

1. \(x = -3 + t, \, y = 5 + t, \, z = 2 + t.\)

Now, calculate the perpendicular distance from the point \((-2, r, 1)\) to the line. The formula for the distance from a point \((x_1, y_1, z_1)\) to a line passing through \((x_0, y_0, z_0)\) with direction \(\vec{d} = (a, b, c)\) is:

1. \(d = \frac{|(x_1 - x_0)a + (y_1 - y_0)b + (z_1 - z_0)c|}{\sqrt{a^2 + b^2 + c^2}}.\)

Substituting the values, we have:

• \(x_0 = -3, \, y_0 = 5, \, z_0 = 2, \, (a, b, c) = (1, 1, 1)\)
• \(x_1 = -2, \, y_1 = r, \, z_1 = 1\)

Squaring both sides gives:

1. \(\frac{(r - 5)^2}{3} = \frac{14}{3}.\)

Thus, \((r - 5)^2 = 14\). Solving this gives:

1. \(r - 5 = \pm\sqrt{14}.\)

Hence, the possible values for \(r\) are:

1. \(r = 5 + \sqrt{14} \quad \text{and} \quad r = 5 - \sqrt{14}.\)

The sum of these possible values is:

1. \((5 + \sqrt{14}) + (5 - \sqrt{14}) = 10.\)

Therefore, the sum of all possible values of \( r \) is 10.