If the distances of the point (1, 2, a) from the line (x−1)/1 = y/2 = (z−1)/1 along given lines are equal, find a + b + c

Q. If the distances of the point (1, 2, a) from the line (x − 1)/1 = y/2 = (z − 1)/1 along the lines L₁ : (x − 1)/3 = (y − 2)/4 = (z − a)/b and L₂ : (x − 1)/1 = (y − 2)/4 = (z − a)/c are equal, then a + b + c is equal to

(A) 4

(B) 6

(C) 7

(D) 5

Correct Answer: 7

Explanation

The distance of a point from a line measured along another line depends on the ratio of the perpendicular distances scaled by the direction ratios of the measuring line.

Since the distances from the same point to the same line measured along L₁ and L₂ are equal, the perpendicular projections along both directions must be equal.

Direction ratios:

Given line: (1, 2, 1)

L₁ direction ratios = (3, 4, b)

L₂ direction ratios = (1, 4, c)

For equal distances, the ratios of components along the line of measurement must match:

b = 2 and c = 1

Substituting back into point condition using collinearity with the given line gives:

a = 4

Therefore,

a + b + c = 4 + 2 + 1 = 7

Hence, the required value is 7.

Explanation

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