JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Three Dimensional Geometry

WhatsApp Email

(a, b, c)

is the image of the point

(1, 2, - 3)

in the line,

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

, then

a + b + c

is equal to

(A)3
(B)1
(C)2✓ Correct
(D)-1

Answer: (C)

Solution

Write a general point of the line as

R = (- 1 + 2 r, 3 - 2 r, - r)

. For

R

to be the foot of the perpendicular from

P = (1, 2, - 3)

, the vector

P R = (2 r - 2, 1 - 2 r, 3 - r)

must be perpendicular to the line's direction

(2, - 2, - 1)

2 (2 r - 2) - 2 (1 - 2 r) - (3 - r) = 9 r - 9 = 0

, so

r = 1

and

R = (1, 1, - 1)

. The image is the reflection of

P

R

, so

(a, b, c) = (2 \cdot 1 - 1, 2 \cdot 1 - 2, 2 (- 1) - (- 3)) = (1, 0, 1)

, and

a + b + c = 2

JEE Main · 2020 · 05 Sep 2020 · Shift 1 · Single correct option

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 →