Let the plane \(\mathrm{P}: \overrightarrow{\mathrm{r}} \cdot \overrightarrow{\mathrm{a}}=\mathrm{d}\) contain the line of intersection of two planes \(\overrightarrow{\mathrm{r}} \cdot(\hat{\mathrm{i}}+3 \hat{\mathrm{j}}-\hat{\mathrm{k}})=6\) and \(\overrightarrow{\mathrm{r}} \cdot(-6 \hat{\mathrm{i}}+5 \hat{\mathrm{j}}-\hat{\mathrm{k}})=7\). If the plane \(P\) passes through the point \(\left(2,3, \frac{1}{2}\right)\), then the value of \(\frac{|13 \overrightarrow{\mathrm{a}}|^{2}}{\mathrm{~d}^{2}}\) is equal to

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Three Dimensional Geometry

Let the plane

P : r \cdot a = d

contain the line of intersection of two planes

r \cdot (\hat{i} + 3 \hat{j} - \hat{k}) = 6

and

r \cdot (- 6 \hat{i} + 5 \hat{j} - \hat{k}) = 7

. If the plane

P

passes through the point

(2, 3, \frac{1}{2})

, then the value of

\frac{∣13 a ∣ ^{2}}{d ^{2}}

is equal to

Answer: (B)

The plane through the intersection is

(r \cdot (\hat{i} + 3 \hat{j} - \hat{k}) - 6) + λ (r \cdot (- 6 \hat{i} + 5 \hat{j} - \hat{k}) - 7) = 0

.

Substituting

(2, 3, \frac{1}{2})

(2 + 9 - \frac{1}{2} - 6) + λ (- 12 + 15 - \frac{1}{2} - 7) = 0

, giving

λ = 1

.

The plane equation becomes

r \cdot (- 5 \hat{i} + 8 \hat{j} - 2 \hat{k}) = 13

, so

a = - 5 \hat{i} + 8 \hat{j} - 2 \hat{k}

and

d = 13

.

Then

∣ a ∣^{2} = 25 + 64 + 4 = 93

, and

\frac{∣13 a ∣ ^{2}}{d ^{2}} = \frac{169 \cdot 93}{169} = 93

JEE Main · 2022 · 28 Jun 2022 · Shift 1 · Single correct option