The line (x - 1/λ )=(y/2)=(z + 3/- 12) makes an isosceles triangle with the planes x+2y-3z+5=0 and 2x+3y-z+2=0, then value of λ can be

Question

The line (x - 1/λ )=(y/2)=(z + 3/- 12) makes an isosceles triangle with the planes x+2y-3z+5=0 and 2x+3y-z+2=0, then value of λ can be (A) 5 (B) 11 (C) 22 (D) 9

Tardigrade Admin · Accepted Answer

Line makes equal angles with both planes and hence equal angles with normals of both planes (λ + 4 + 36/√λ 2 + 4 + 144 √12 + 22 + 32)=(2 λ + 6 + 12/√λ 2 + 4 + 144 √12 + 22 + 32)∴ λ =22

Q. The line $\frac{x - 1}{λ} = \frac{y}{2} = \frac{z + 3}{- 12}$ makes an isosceles triangle with the planes $x + 2 y - 3 z + 5 = 0$ and $2 x + 3 y - z + 2 = 0,$ then value of $λ$ can be

$5$

$11$

$22$

$9$

Line makes equal angles with both planes and hence equal angles with normals of both planes
$\frac{λ + 4 + 36}{λ ^{2} + 4 + 144 1 ^{2} + 2 ^{2} + 3 ^{2}} = \frac{2 λ + 6 + 12}{λ ^{2} + 4 + 144 1 ^{2} + 2 ^{2} + 3 ^{2}} ∴ λ = 22$

Q. The line λx−1​=2y​=−12z+3​ makes an isosceles triangle with the planes x+2y−3z+5=0 and 2x+3y−z+2=0, then value of λ can be

5

11

22

9

Line makes equal angles with both planes and hence equal angles with normals of both planes λ2+4+144​12+22+32​λ+4+36​=λ2+4+144​12+22+32​2λ+6+12​∴λ=22

Q. The line $\frac{x - 1}{λ} = \frac{y}{2} = \frac{z + 3}{- 12}$ makes an isosceles triangle with the planes $x + 2 y - 3 z + 5 = 0$ and $2 x + 3 y - z + 2 = 0,$ then value of $λ$ can be

$5$

$11$

$22$

$9$

Line makes equal angles with both planes and hence equal angles with normals of both planes
$\frac{λ + 4 + 36}{λ ^{2} + 4 + 144 1 ^{2} + 2 ^{2} + 3 ^{2}} = \frac{2 λ + 6 + 12}{λ ^{2} + 4 + 144 1 ^{2} + 2 ^{2} + 3 ^{2}} ∴ λ = 22$