Question

8.
The planes 2x - y + 4z = 5 and
5x – 2.5y + 10z = 6 are
(A) Parallel to each other
(B) Pass through (0, 0, 54
(C) Perpendicular to each other
(D) Intersect y-axis

Answer 1

Answer:

(A) Parallel to each other

Step-by-step explanation:

We know that if two planes are parallel than ratio of normal's direction ratios are equal.

$\frac{a_1}{a_2} = \frac{b_1}{b_2} = \frac{c_1}{c_2}$

Planes are -2x + y - 4z+ 5=0 and

-5x + 2.5y - 10z + 6 =0

For plane 1: a1= -2, b1=1,c1=-4

For plane 2: a2= -5, b2=2.5,c2=-10

$\frac{ - 2}{ - 5} = \frac{1}{2.5} = \frac{4}{10} \\ \\ \frac{ - 2}{ - 5} = \frac{10}{25} = \frac{4}{10} \\ \\ \frac{ 2}{ 5} = \frac{2}{5} = \frac{2}{5}$

Hence both planes are parallel.

Option A is correct.

Hope it helps you.

Answer 2

Formula:

Let the two planes be

a₁x + b₁y + c₁z + d₁ = 0

and a₂x + b₂y + c₂z + d₂ = 0

If we take θ as the angle between the two planes, then θ is the angle between their normals. θ is determined by

cosθ = (a₁a₁ + b₁b₂ + c₁c₂)/{√(a₁² + b₁² + c₁²).√(a₂² + b₂² + c₂²)}

Corollary 1.

If the two planes be perpendicular, then

a₁a₂ + b₁b₂ + c₁c₂ = 0

Corollary 2.

If the two planes be parallel to each other, then

a₁/a₂ = b₁/b₂ = c₁/c₂

Solution:

The two given planes are

2x - y + 4z = 5 ..... (1)

5x - 2.5y + 10z = 6 ..... (2)

• Now,

2 * 5 + (- 1) * (- 2.5) + 4 * 10 = 52.5 ≠ 0

So (1) and (2) planes aren't perpendicular to each other.

• Now, 2 / 5 = 0.4

(- 1) / (- 2.5) = 0.4

4 / 10 = 0.4

So, 2 / 5 = (- 1) / (- 2.5) = 4 / 10

and thus the two planes are parallel to each other.

• The point (0, 0, 5) doesn't not satisfy any of the planes (1) and (2), and thus (0, 0, 5) doesn't lie on them.

• Also the planes do not intersect the y-axis (x = 0).

Therefore, option (A) is correct.