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MCQMediumJEE 2026Equation of Line in 3D

JEE Mathematics 2026 Question with Solution

If the distances of the point (1,2,a)(1,2,a) from the line x−11=y2=z−11\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1} along the lines L1: x−13=y−24=z−abL_1:\ \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} and L2: x−11=y−24=z−acL_2:\ \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c} are equal, then a+b+ca+b+c is equal to:

  • A

    55

  • B

    66

  • C

    44

  • D

    77

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The point is (1,2,a)(1,2,a). The given line is

x−11=y2=z−11\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}

and the two lines are

L1: x−13=y−24=z−abL_1:\ \frac{x-1}{3}=\frac{y-2}{4}=\frac{z-a}{b} L2: x−11=y−24=z−acL_2:\ \frac{x-1}{1}=\frac{y-2}{4}=\frac{z-a}{c}

Find: a+b+ca+b+c when the distances from the point to the given line measured along L1L_1 and L2L_2 are equal.

Step 1: Direction ratios from the solution are:

(1,2,1)(1,2,1)

for the given line,

(3,4,b)(3,4,b)

for L1L_1, and

(1,4,c)(1,4,c)

for L2L_2.

Step 2: Using the condition for equal distances measured along the two lines, the solution gives

a−1b=a−1c\frac{a-1}{b}=\frac{a-1}{c}

Hence,

b=cb=c

Step 3: Using the coplanarity/proportionality condition stated in the solution,

31=42=b1\frac{3}{1}=\frac{4}{2}=\frac{b}{1}

So,

b=3b=3

Therefore,

c=3c=3

Step 4: From the given line,

x−11=y2=z−11\frac{x-1}{1}=\frac{y}{2}=\frac{z-1}{1}

the solution concludes

a=1a=1

Step 5: Therefore,

a+b+c=1+3+1=5a+b+c=1+3+1=5

So the correct option is A.

Extracted Hint and Key Idea

Hint from the solution: When distances are measured along lines, compare ratios of direction ratios instead of using perpendicular distance formulas.

The extracted solution identifies the relevant direction ratios, uses the equality condition to obtain b=cb=c, then applies the stated proportionality condition to get b=3b=3 and concludes the required sum is 55. Hence, the correct option is A.

Common mistakes

  • Using the perpendicular distance formula from a point to a line is incorrect here because the distance is measured along given lines L1L_1 and L2L_2. Instead, use the directional condition described in the solution.

  • Ignoring direction ratios of the lines can lead to a wrong setup. First extract the direction ratios of the given line, L1L_1, and L2L_2 correctly before applying the equality condition.

  • After obtaining b=cb=c, concluding the answer immediately is wrong because aa must still be found and the actual values of bb and cc must be determined using the additional condition stated in the solution.

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