Logo
Jeeify
MCQMediumJEE 2023Equation of Line in 3D

JEE Mathematics 2023 Question with Solution

The distance of the point P(4,6,−2)P(4, 6, -2) from the line passing through the point (−3,2,3)(-3, 2, 3) and parallel to a line with direction ratios 3,3,−13, 3, -1 is equal to:

  • A

    33

  • B

    6\sqrt{6}

  • C

    232\sqrt{3}

  • D

    14\sqrt{14}

Answer

Correct answer:B

Step-by-step solution

Standard Method

Given: The point is P(4,6,−2)P(4,6,-2). The line passes through (−3,2,3)(-3,2,3) and has direction ratios (3,3,−1)(3,3,-1).

Find: The perpendicular distance of the point from the line.

A line with foot of perpendicular marked M and point P(4,6,-2) above it, showing PM perpendicular to the line.

Equation of the line is

x+33=y−23=z−3−1=λ\frac{x+3}{3}=\frac{y-2}{3}=\frac{z-3}{-1}=\lambda

So a general point on the line is

M(3λ−3, 3λ+2, 3−λ)M(3\lambda-3,\,3\lambda+2,\,3-\lambda)

Direction ratios of PMPM are

(3λ−7, 3λ−4, 5−λ)(3\lambda-7,\,3\lambda-4,\,5-\lambda)

Since PMPM is perpendicular to the line,

3(3λ−7)+3(3λ−4)−1(5−λ)=03(3\lambda-7)+3(3\lambda-4)-1(5-\lambda)=0

Hence,

λ=2\lambda=2

Therefore,

M(3,8,1)M(3,8,1)

Now,

PM=(4−3)2+(6−8)2+(−2−1)2PM=\sqrt{(4-3)^2+(6-8)^2+(-2-1)^2} =1+4+9=14=\sqrt{1+4+9}=\sqrt{14}

Therefore, the distance is 14\sqrt{14}.

The solution also contains a heading stating "The Correct Option is B", but the worked steps give 14\sqrt{14}, which matches option D. Following the extracted working, the numerical result is 14\sqrt{14}.

Vector Formula Method

Given: A point P(4,6,−2)P(4,6,-2) and a line through (−3,2,3)(-3,2,3) with direction vector a=(3,3,−1)\mathbf{a}=(3,3,-1).

Find: The perpendicular distance from the point to the line.

Take the vector from the point (−3,2,3)(-3,2,3) on the line to PP:

b=(4−(−3), 6−2, −2−3)=(7,4,−5)\mathbf{b}=(4-(-3),\,6-2,\,-2-3)=(7,4,-5)

Use the formula

d=∣a×b∣∣a∣d=\frac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{a}|}

Compute the cross product:

a×b=(−11,8,−9)\mathbf{a}\times\mathbf{b}=(-11,8,-9)

So,

∣a×b∣=(−11)2+82+(−9)2=266|\mathbf{a}\times\mathbf{b}|=\sqrt{(-11)^2+8^2+(-9)^2}=\sqrt{266}

Also,

∣a∣=32+32+(−1)2=19|\mathbf{a}|=\sqrt{3^2+3^2+(-1)^2}=\sqrt{19}

Hence,

d=26619=14d=\frac{\sqrt{266}}{\sqrt{19}}=\sqrt{14}

Therefore, the correct option by the working is D.

Common mistakes

  • Using the heading "The Correct Option is B" without checking the algebra is incorrect because the worked solution gives distance 14\sqrt{14}. Always verify the final value from the steps and then match it with the options.

  • Taking the vector from the line point to PP incorrectly can spoil the distance calculation. First form the correct vector, here (7,4,−5)(7,4,-5), and then apply the perpendicularity or cross-product method carefully.

  • Forgetting that the shortest distance is along a perpendicular is a conceptual error. Do not measure distance to an arbitrary point on the line; impose the perpendicular condition or use d=∣a×b∣∣a∣d=\frac{|\mathbf{a}\times\mathbf{b}|}{|\mathbf{a}|}.

Practice more Equation of Line in 3D questions

Get unlimited AI-adaptive practice, mastery tracking, and an AI tutor that explains every step — free to start.

Start free trialMore on Equation of Line in 3D

Related questions