Let the mirror image of the point $(1,3,a)$ with respect to the plane $\vec{r} \cdot(2 \hat{i}-\hat{j}+\hat{k})-b=0$ be $(-3,5,2)$. Then, the value of $|a+b|$ is equal to

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Three Dimensional Geometry

Let the mirror image of the point

(1, 3, a)

with respect to the plane

r \cdot (2 \hat{i} - \hat{j} + \hat{k}) - b = 0

(- 3, 5, 2)

. Then, the value of

∣ a + b ∣

is equal to

Answer: 1

Solution

The plane is

2 x - y + z = b

, with normal

(2, - 1, 1)

. The segment joining the point

(1, 3, a)

to its image

(- 3, 5, 2)

runs along this normal, so

\frac{- 3 - 1}{2} = \frac{5 - 3}{- 1} = \frac{2 - a}{1}

The first two ratios are both

- 2

, so

2 - a = - 2

and

a = 4

. The midpoint of the segment,

(- 1, 4, 3)

, lies on the plane, hence

b = 2 (- 1) - 4 + 3 = - 3

Therefore

∣ a + b ∣ = ∣4 - 3∣ = 1

JEE Main · 2021 · 18 Mar 2021 · Shift 2 · Integer answer

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