JEE Main 202313 AprilEvening Shift

Exam: JEE Main
Year: 2023
Date: 13 April
Shift: Evening Shift
Subject: Mathematics
Chapter: Three Dimensional Geometry

JEE Main 2023 · 13 April · Evening Shift · Mathematics · Three Dimensional Geometry

The plane, passing through the points ( 0, – 1, 2 ) and ( – 1, 2, 1...

The plane, passing through the points

( 0, – 1, 2 )

and

( – 1, 2, 1 )

and parallel to the line passing through

( 5, 1, – 7 )

and

( 1, – 1, – 1 ) ,

also passes through the point

Options

A $\left(- 2, 5, 0\right)$
B $\left(1, - 2, 1\right)$
C $\left(2, 0, 1\right)$
D $\left(0, 5, - 2\right)$

Correct answer

A. $\left(- 2, 5, 0\right)$

Step-by-step solution

Let $A\left(0, - 1, 2\right) \text{ and } B\left(- 1, 2, 1\right)$ and given plane is parallel to line passing through \left(5, 1, - 7\right) \text{ and } \left(1, - 1, - 1\right) $So, normal vector to the plane is given by,$ \overset{\to}{n} = \left| $\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ - 1 - 0 & 2 + 1 & 1 - 2 \\ 1 - 5 & - 1 - 1 & - 1 + 7\end{cases}$ \right| $$ ⇒ \overset{\to}{n} = \left| $\begin{cases}\overset{^}{i} & \overset{^}{j} & \overset{^}{k} \\ - 1 & 3 & - 1 \\ - 4 & - 2 & 6\end{cases}$ \right| = \overset{^}{i}\left(16\right) - \overset{^}{j}\left(- 10\right) + \overset{^}{k}\left(14\right) $$ ⇒ \overset{\to}{n} = 16\overset{^}{i} + 10\overset{^}{j} + 14\overset{^}{k} $$ ∴ $Equation of plane$ 16 ( x – 0 ) + 10 ( y + 1 ) + 14 ( z – 2 ) = 0 $$ ⇒ 8x + 5y + 7z = 9 $Now satisfying the points from option in the plane we get, \left(- 2, 5, 0\right)$ only point which satisfy the plane.

Practice Three Dimensional Geometry on Quantrex Academy →

More from Three Dimensional Geometry

Let P be the plane such that it contains the straight line (x-1)/(2) = (y-3)/(3) =… JEE Advanced 2026 Let L be the straight line joining the points P(1, 2, -1) and Q(2, 3, 1) . Let S be the… JEE Advanced 2026 Let the foot of perpendicular from the point (λ, 2, 3) on the line (x-4)/(1) = (y-9)/(2)… JEE Main 2026 Let a line L_1 pass through the origin and be perpendicular to the lines L_2: r = (3+t) i… JEE Main 2026 The shortest distance between the lines (x-4)/(1) = (y-3)/(2) = (z-2)/(-3) and (x+2)/(2)… JEE Main 2026 Let the image of the point P(0, -5, 0) in the line (x-1)/(2) = (y)/(1) = (z+1)/(-2) be… JEE Main 2026 Let the image of the point P(1, 6, a) in the line L: (x)/(1) = (y-1)/(2) = (z-a+1)/(b)… JEE Main 2026 Let a line L be perpendicular to both the lines L_1: (x+1)/(3) = (y+3)/(5) = (z+5)/(7)… JEE Main 2026 Let a triangle PQR be such that P and Q lie on the line (x+3)/(8) = (y-4)/(2) = (z+1)/(2)… JEE Main 2026 If the distance of the point (a, 2, 5) from the image of the point (1, 2, 7) in the line… JEE Main 2026 The square of the distance of the point P(5, 6, 7) from the line (x-2)/(2) = (y-5)/(3) =… JEE Main 2026 The square of the distance of the point of intersection of the lines r = ( i + j - k ) +… JEE Main 2026 The shortest distance between the lines r =((1)/(3) i +2 j +(8)/(3) k )+λ(2 i -5 j +6 k )… JEE Main 2026 If (2α+1, α^2-3α, (α-1)/(2)) is the image of (α, 2α, 1) in the line… JEE Main 2026 A line with direction ratios 1, -1, 2 intersects the lines (x)/(2) = (y)/(3) = (z+1)/(3)… JEE Main 2026 The square of the distance of the point (-2, -8, 6) from the line (x-1)/(1) = (y-1)/(2) =… JEE Main 2026 Full Three Dimensional Geometry list Three Dimensional Geometry JEE Maths PYQs JEE Mathematics PYQs All JEE Main PYQs