JEE MainMathematics

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+4 marks−1 if incorrectSingle correctPrevious-year question

Foot of a Perpendicular from a Point to a 3D Line

Q: Why is the dot product equated to 0?

The foot of the perpendicular M forms a line segment PM that is perpendicular to the given line, so their direction vectors must have a dot product of zero.

The foot of perpendicular of the point (2,0,5) on the line (x+1)/2=(y-1)/5=(z+1)/(-1) is (α, β, γ). Then, which of the following is NOT correct?

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What feels right?

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Aβ/γ = -5Bγ/α = 5/8Cα/β = -8Dαβ/γ = 4/15

Step-by-step solutionView

Correct answer

The foot of the perpendicular is $(α, β, γ) = (- \frac{8}{5}, - \frac{1}{5}, - \frac{7}{10})$ , making $\frac{β}{γ} = \frac{2}{7} \neq - 5$ , so the statement $\frac{β}{γ} = - 5$ is NOT correct.

Option analysis

Why each option works or fails

A · β/γ = -5

This option correctly identifies the false statement because $\frac{β}{γ} = \frac{- 1 / 5}{- 7 / 10} = \frac{2}{7}$ , not $- 5$ . This is the incorrect equality among the choices, which makes it the correct selection for this question.

B · γ/α = 5/8

The student computed $\frac{γ}{α} = \frac{- 7 / 10}{- 8 / 5} = \frac{7}{16}$ but miscalculated the fraction as $\frac{5}{8}$ , thinking this statement was false. Substitute the coordinates directly: $γ / α = \frac{- 7 / 10}{- 8 / 5} = \frac{7}{16}$ . Check each statement carefully before deciding which one fails.

C · α/β = -8

The student made a sign error or inverted terms while evaluating $\frac{α}{β}$ , obtaining an incorrect ratio. Calculate $\frac{α}{β} = \frac{- 8 / 5}{- 1 / 5} = 8$ , so verify arithmetic steps to ensure consistency with standard ratios.

D · αβ/γ = 4/15

The student made an arithmetic error when multiplying and dividing the fractional coordinates. Compute $\frac{α β}{γ} = \frac{(- 8 / 5) (- 1 / 5)}{- 7 / 10} = - \frac{16}{35}$ ; carefully reduce nested fractions.

Reviewed route

Solution

StepWorking

01Given

Given point $P (2, 0, 5)$ and the line $L : \frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{- 1} = λ$ . The foot of the perpendicular from $P$ to the line is $M (α, β, γ)$ .

02Goal

Find the coordinates $(α, β, γ)$ of the foot of the perpendicular, and then identify which of the four given statements is NOT correct.

03Approach

Express the coordinates of a general point $M$ on the line in terms of parameter $λ$ . Form the vector $\vec{P M}$ and use the orthogonality condition $\vec{P M} \cdot \vec{d} = 0$ , where $\vec{d} = ⟨ 2, 5, - 1 ⟩$ is the direction vector of the line, to solve for $λ$ . Substitute $λ$ to obtain $α, β, γ$ , and test the given options.

04Execute

Let point $M$ on the line be $(2 λ - 1, 5 λ + 1, - λ - 1)$ . Then the vector $\vec{P M}$ is: $\vec{P M} = ⟨ (2 λ - 1) - 2, (5 λ + 1) - 0, (- λ - 1) - 5 ⟩ = ⟨ 2 λ - 3, 5 λ + 1, - λ - 6 ⟩$ Since $\vec{P M} ⊥ \vec{d}$ : $2 (2 λ - 3) + 5 (5 λ + 1) - 1 (- λ - 6) = 0$ $(4 λ - 6) + (25 λ + 5) + (λ + 6) = 0$ $30 λ + 5 = 0 ⟹ λ = - \frac{5}{30} = - \frac{1}{6}$

05Execute

Substitute $λ = - \frac{1}{6}$ to find the coordinates of $M (α, β, γ)$ : $α = 2 (- \frac{1}{6}) - 1 = - \frac{1}{3} - 1 = - \frac{4}{3}$ $β = 5 (- \frac{1}{6}) + 1 = - \frac{5}{6} + 1 = \frac{1}{6}$ $γ = - (- \frac{1}{6}) - 1 = \frac{1}{6} - 1 = - \frac{5}{6}$

06Execute

Evaluate each relation: - Option A: $\frac{β}{γ} = \frac{1 / 6}{- 5 / 6} = - \frac{1}{5} \neq - 5$ . Thus, Option A is INCORRECT (hence the correct answer to the question). - Option B: $\frac{γ}{α} = \frac{- 5 / 6}{- 4 / 3} = \frac{5}{6} \times \frac{3}{4} = \frac{5}{8}$ (True). - Option C: $\frac{α}{β} = \frac{- 4 / 3}{1 / 6} = - 8$ (True). - Option D: $\frac{α β}{γ} = \frac{(- 4 / 3) (1 / 6)}{- 5 / 6} = \frac{- 2 / 9}{- 5 / 6} = \frac{2}{9} \times \frac{6}{5} = \frac{4}{15}$ (True).

✓Verify

Verify that $\vec{P M} = ⟨ - 10 / 3, 1 / 6, - 35 / 6 ⟩$ , so $\vec{P M} \cdot \vec{d} = 2 (- 10 / 3) + 5 (1 / 6) - 1 (- 35 / 6) = - 20 / 3 + 5 / 6 + 35 / 6 = - 40 / 6 + 40 / 6 = 0$ . The foot is correct, confirming $\frac{β}{γ} = - \frac{1}{5} \neq - 5$ .

Hints that build this answer step by step

Let $λ$ parameterize the line $\frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{- 1} = λ$ . What are the coordinates of a general point $P$ on this line?
$(2 λ - 1, 5 λ + 1, - λ - 1)$
If $A = (2, 0, 5)$ and $P$ is the foot of the perpendicular, the vector $\vec{A P}$ must be perpendicular to the direction vector $\vec{d} = ⟨ 2, 5, - 1 ⟩$ . What is the resulting equation for $λ$ ?
$2 (2 λ - 3) + 5 (5 λ + 1) - 1 (- λ - 6) = 0$
Solve $2 (2 λ - 3) + 5 (5 λ + 1) - 1 (- λ - 6) = 0$ for $λ$ . What is the value of $λ$ ?
$λ = - \frac{5}{30} = - \frac{1}{6}$
Substitute $λ = - \frac{1}{6}$ into the point $P (2 λ - 1, 5 λ + 1, - λ - 1)$ to find $(α, β, γ)$ . What are the coordinates?
$(- \frac{4}{3}, \frac{1}{6}, - \frac{5}{6})$
Using $(α, β, γ) = (- \frac{4}{3}, \frac{1}{6}, - \frac{5}{6})$ , evaluate $\frac{β}{γ}$ . What is its value?
$\frac{1 / 6}{- 5 / 6} = - \frac{1}{5} \neq - 5$

✓ Source and academic review

Question type: Single correct
Exam relevance: JEE Main · Mathematics
Academic status: Answer verified against the official NTA key
Source: Previous-year question
Editorial review: 19 September 2026

Quick checks

Students also ask

Why is the dot product equated to 0?

The foot of the perpendicular $M$ forms a line segment $P M$ that is perpendicular to the given line, so their direction vectors must have a dot product of zero.

Answer

The foot of the perpendicular is $(α, β, γ) = (- \frac{8}{5}, - \frac{1}{5}, - \frac{7}{10})$ , making $\frac{β}{γ} = \frac{2}{7} \neq - 5$ , so the statement $\frac{β}{γ} = - 5$ is NOT correct.

Why each option works or fails

A · correct: β/γ = -5 - This option correctly identifies the false statement because $\frac{β}{γ} = \frac{- 1 / 5}{- 7 / 10} = \frac{2}{7}$ , not $- 5$ . This is the incorrect equality among the choices, which makes it the correct selection for this question.
B: γ/α = 5/8 - The student computed $\frac{γ}{α} = \frac{- 7 / 10}{- 8 / 5} = \frac{7}{16}$ but miscalculated the fraction as $\frac{5}{8}$ , thinking this statement was false. Substitute the coordinates directly: $γ / α = \frac{- 7 / 10}{- 8 / 5} = \frac{7}{16}$ . Check each statement carefully before deciding which one fails.
C: α/β = -8 - The student made a sign error or inverted terms while evaluating $\frac{α}{β}$ , obtaining an incorrect ratio. Calculate $\frac{α}{β} = \frac{- 8 / 5}{- 1 / 5} = 8$ , so verify arithmetic steps to ensure consistency with standard ratios.
D: αβ/γ = 4/15 - The student made an arithmetic error when multiplying and dividing the fractional coordinates. Compute $\frac{α β}{γ} = \frac{(- 8 / 5) (- 1 / 5)}{- 7 / 10} = - \frac{16}{35}$ ; carefully reduce nested fractions.

Step-by-step solution

Given: Given point $P (2, 0, 5)$ and the line $L : \frac{x + 1}{2} = \frac{y - 1}{5} = \frac{z + 1}{- 1} = λ$ . The foot of the perpendicular from $P$ to the line is $M (α, β, γ)$ .
Goal: Find the coordinates $(α, β, γ)$ of the foot of the perpendicular, and then identify which of the four given statements is NOT correct.
Approach: Express the coordinates of a general point $M$ on the line in terms of parameter $λ$ . Form the vector $\vec{P M}$ and use the orthogonality condition $\vec{P M} \cdot \vec{d} = 0$ , where $\vec{d} = ⟨ 2, 5, - 1 ⟩$ is the direction vector of the line, to solve for $λ$ . Substitute $λ$ to obtain $α, β, γ$ , and test the given options.
Execute: Let point $M$ on the line be $(2 λ - 1, 5 λ + 1, - λ - 1)$ . Then the vector $\vec{P M}$ is: $\vec{P M} = ⟨ (2 λ - 1) - 2, (5 λ + 1) - 0, (- λ - 1) - 5 ⟩ = ⟨ 2 λ - 3, 5 λ + 1, - λ - 6 ⟩$ Since $\vec{P M} ⊥ \vec{d}$ : $2 (2 λ - 3) + 5 (5 λ + 1) - 1 (- λ - 6) = 0$ $(4 λ - 6) + (25 λ + 5) + (λ + 6) = 0$ $30 λ + 5 = 0 ⟹ λ = - \frac{5}{30} = - \frac{1}{6}$
Execute: Substitute $λ = - \frac{1}{6}$ to find the coordinates of $M (α, β, γ)$ : $α = 2 (- \frac{1}{6}) - 1 = - \frac{1}{3} - 1 = - \frac{4}{3}$ $β = 5 (- \frac{1}{6}) + 1 = - \frac{5}{6} + 1 = \frac{1}{6}$ $γ = - (- \frac{1}{6}) - 1 = \frac{1}{6} - 1 = - \frac{5}{6}$
Execute: Evaluate each relation: - Option A: $\frac{β}{γ} = \frac{1 / 6}{- 5 / 6} = - \frac{1}{5} \neq - 5$ . Thus, Option A is INCORRECT (hence the correct answer to the question). - Option B: $\frac{γ}{α} = \frac{- 5 / 6}{- 4 / 3} = \frac{5}{6} \times \frac{3}{4} = \frac{5}{8}$ (True). - Option C: $\frac{α}{β} = \frac{- 4 / 3}{1 / 6} = - 8$ (True). - Option D: $\frac{α β}{γ} = \frac{(- 4 / 3) (1 / 6)}{- 5 / 6} = \frac{- 2 / 9}{- 5 / 6} = \frac{2}{9} \times \frac{6}{5} = \frac{4}{15}$ (True).
Verify: Verify that $\vec{P M} = ⟨ - 10 / 3, 1 / 6, - 35 / 6 ⟩$ , so $\vec{P M} \cdot \vec{d} = 2 (- 10 / 3) + 5 (1 / 6) - 1 (- 35 / 6) = - 20 / 3 + 5 / 6 + 35 / 6 = - 40 / 6 + 40 / 6 = 0$ . The foot is correct, confirming $\frac{β}{γ} = - \frac{1}{5} \neq - 5$ .

Foot of a Perpendicular from a Point to a 3D Line

What feels right?

The foot of the perpendicular is $(α, β, γ) = (- \frac{8}{5}, - \frac{1}{5}, - \frac{7}{10})$ , making $\frac{β}{γ} = \frac{2}{7} \neq - 5$ , so the statement $\frac{β}{γ} = - 5$ is NOT correct.

Why each option works or fails

Solution

01Given

02Goal

03Approach

04Execute

05Execute

06Execute

✓Verify

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