JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Three Dimensional Geometry

Let the plane

a x + b y + cz + d = 0

bisect the line joining the points

(4, - 3, 1)

and

(2, 3, - 5)

at the right angles. If

a, b, c, d

are integer, then the minimum value of

(a^{2} + b^{2} + c^{2} + d^{2})

Answer: 28

Solution

The plane is the perpendicular bisector of the segment, so its normal lies along

(2, 3, - 5) - (4, - 3, 1) = (- 2, 6, - 6) ∥ (1, - 3, 3)

and it passes through the midpoint

(3, 0, - 2)

. Its equation is

(x - 3) - 3 (y - 0) + 3 (z + 2) = 0

, that is

x - 3 y + 3 z + 3 = 0

Every integer quadruple

(a, b, c, d)

describing this plane is an integer multiple of

(1, - 3, 3, 3)

, so the smallest value of

a^{2} + b^{2} + c^{2} + d^{2}

comes from the primitive quadruple itself and equals

1 + 9 + 9 + 9 = 28

JEE Main · 2021 · 18 Mar 2021 · Shift 1 · Integer answer

Practise more: all JEE Main PYQs on Three Dimensional Geometry →

More JEE Main questions on Three Dimensional Geometry

Let $(α, β, γ)$ be the co-ordinates of the foot of the perpendicular drawn from the point $(5, 4, 2)$ on the line $r = (- \hat{i} + 3 \hat{j} + \hat{k}) + λ (2 \hat{i} + 3 \hat{j} - \hat{k})$ .
Then the length of the projection of the vector …
JEE Main 2026View solution →
Let the line L pass through the point $(- 3, 5, 2)$ and make equal angles with the positive coordinate axes. If the distance of L from the point $(- 2, r, 1)$ is $\frac{14}{3}$ , then the sum of all possible values of $r$ is :
JEE Main 2026View solution →
Let the line $L_{1}$ be parallel to the vector $- 3 \hat{i} + 2 \hat{j} + 4 \hat{k}$ and pass through the point $(2, 6, 7)$ and the line $L_{2}$ be parallel to the vector $2 \hat{i} + \hat{j} + 3 \hat{k}$ and pass through the point $(4, 3, 5)$ . If the line $L_{3}$ is parallel to the vector …
JEE Main 2026View solution →
Let $P (α, β, γ)$ be the point on the line
$\frac{x - 1}{2} = \frac{y + 1}{- 3} = z$ at a distance $414$ from the point $(1, - 1, 0)$ and nearer to the origin. Then the shortest distance, between the lines …
JEE Main 2026View solution →
If the image of the point $P (1, 2, a)$ in the line $\frac{x - 6}{3} = \frac{y - 7}{2} = \frac{7 - z}{2}$ is $Q (5, b, c)$ , then $a^{2} + b^{2} + c^{2}$ is equal to
JEE Main 2026View solution →
Let L be the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z + 3}{6}$ and let S be the set of all points $(a, b, c)$ on L, whose distance from the line $\frac{x + 1}{2} = \frac{y + 1}{3} = \frac{z - 9}{0}$ along the line L is 7. Then $\sum_{(a, b, c) \in S} (a + b + c)$ is equal to :
JEE Main 2026View solution →