Consider the line L passing through the points (1, 2, 3) and (2, 3, 5). The distance of the point ( 113, 113, 193) from the line L along the line:

Given: The line passes through (1,2,3) and (2,3,5). The point is A ( 113, 113, 193). Find: The distance of point A from the given line along a line parallel to the required direction. The equation of the given line is x-12-1= y-23-2= z-35-3 So, x-11= y-21= z-32= Hence, a general point on the line is B(1+, 2+, 3+2) and the direction ratios of the line are 1,1,2. Now the direction ratios of line AB are 3 -83, 3 -53, 6 -103 Since AB is taken along the required line, it is parallel to the given direction. Therefore, 3 -83 -5= 21 This gives 3 -8=6 -10 so 3 =2 and hence = 23 Substitute = 23 in B=(1+, 2+, 3+2) to get B= ( 53, 83, 133) Now, AB= ( 113- 53)^2+ ( 113- 83)^2+ ( 193- 133)^2 = ( 63)^2+ ( 33)^2+ ( 63)^2 = 2^2+1^2+2^2= 9=3 Therefore, the distance is 3 and the correct option is A.

Consider the line L passing through the points (1, 2, 3) and (2,… — Solution | JEE Mathematics

MCQMediumJEE 2024Equation of Line in 3D

Consider the line $L$ passing through the points $(1, 2, 3)$ and $(2, 3, 5)$ . The distance of the point $\left(\frac{11}{3}, \frac{11}{3}, \frac{19}{3}\right)$ from the line $L$ along the line:

A
$3$
B
$5$
C
$4$
D
$6$

Answer

Correct answer:A

Step-by-step solution

Standard Method

Given: The line passes through $(1,2,3)$ and $(2,3,5)$ . The point is $A\left(\frac{11}{3},\frac{11}{3},\frac{19}{3}\right)$ .

Find: The distance of point $A$ from the given line along a line parallel to the required direction.

The equation of the given line is

\frac{x-1}{2-1}=\frac{y-2}{3-2}=\frac{z-3}{5-3}

So,

\frac{x-1}{1}=\frac{y-2}{1}=\frac{z-3}{2}=\lambda

Hence, a general point on the line is

B(1+\lambda,\;2+\lambda,\;3+2\lambda)

and the direction ratios of the line are $\langle 1,1,2 \rangle$ .

Now the direction ratios of line $AB$ are

\left\langle \frac{3\lambda-8}{3},\;\frac{3\lambda-5}{3},\;\frac{6\lambda-10}{3} \right\rangle

Since $AB$ is taken along the required line, it is parallel to the given direction. Therefore,

\frac{3\lambda-8}{3\lambda-5}=\frac{2}{1}

This gives

3\lambda-8=6\lambda-10

3\lambda=2

and hence

\lambda=\frac{2}{3}

Distance Computation

Substitute $\lambda=\frac{2}{3}$ in

B=(1+\lambda,\;2+\lambda,\;3+2\lambda)

to get

B=\left(\frac{5}{3},\frac{8}{3},\frac{13}{3}\right)

Final Distance

Now,

AB=\sqrt{\left(\frac{11}{3}-\frac{5}{3}\right)^2+\left(\frac{11}{3}-\frac{8}{3}\right)^2+\left(\frac{19}{3}-\frac{13}{3}\right)^2}

=\sqrt{\left(\frac{6}{3}\right)^2+\left(\frac{3}{3}\right)^2+\left(\frac{6}{3}\right)^2}

=\sqrt{2^2+1^2+2^2}=\sqrt{9}=3

Therefore, the distance is $3$ and the correct option is A.

Common mistakes

Using the perpendicular distance formula is incorrect here because the question asks for distance along a line, not the shortest distance. First impose the required parallel condition, then compute the segment length.
Taking wrong direction ratios from the two given points leads to an incorrect point on the line. The correct direction ratios are obtained from $(2-1,\;3-2,\;5-3)=(1,1,2)$ .
Forming point $B$ incorrectly as $(1+\lambda,2+\lambda,3+\lambda)$ is wrong because the third direction component is $2$ , so the third coordinate must be $3+2\lambda$ .

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