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Three Dimensional Geometry

Multiple Choice Questions

471.

The equation of the plane which is equidistant from the two parallel planes 2x - 2y + z + 3= 0 and 4x - 4y + 2z + 9 = 0 is

8x - 8y + 4z + 15 = 0
8x - 8y + 4z - 15 = 0
8x - 8y + 4z + 3 = 0
8x - 8y + 4z - 3 = 0

472.

The angle between the planes 3x + 4y + 5z = 3 and 4x - 3y + 5z = 9 is equal to

$\frac{π}{2}$
$\frac{π}{4}$
$\frac{π}{6}$
$\frac{π}{3}$

473.

The vector equation of the plane through the point (2, 1, - 1) and parallel to the plane r - (i + 3j - k) = 0 is

r . (i + 9j + 11k) = 6
r . (i - 9j + 11k) = 4
r . (i + 3j - k) = 6
r . (i + 3j - k) = 4

474.

If the foot of the perpendicular drawn from the point (5, 1, - 3) to a plane is (1, - 1, 3), then the equation of the plane is

2x + y - 3z + 8 = 0
2x + y + 3z + 8 = 0
2x - y - 3z + 8 = 0
2x - y + 3z + 8 = 0

475.

The equation of the plane through the line of intersection of the planes x - y + z + 3 = 0 and x + y + 22 + 1 = 0 and parallel to x-axis is

2y - z = 2
2y + z = 2
4y + z = 4
y - 2z = 3

476.
If 3p + 2q = i + j + k and 3p - 2q = i - j - k, then the angle between p and q is

$\frac{π}{6}$

$\frac{π}{4}$

$\frac{π}{3}$

$\frac{π}{2}$

$\frac{π}{2}$

Given, 3p + 2q = i + j + k ...(i)

and 3p - 2q = i - j - k ...(ii)

On adding Eqs. (i) and (ii), we get

6p = 2i

$∴ p = \frac{i}{3}$

On subtracting Eq. (ii) from Eq. (i), we get

4q = 2(j + k) $\Rightarrow q = \frac{1}{2} (j + k)$

Let $θ$ be the angle between p and q, then

$p . q = |p| |q| \cos (θ) \Rightarrow \cos (θ) = \frac{p . q}{|p| |q|} \Rightarrow \cos (θ) = \frac{1}{6} \frac{i . (j + k)}{|p| |q|} = 0 = \cos (90 °) ∴ θ = 90 ° = \frac{π}{2}$

477.

The point of intersection of the straight line $\frac{x - 2}{2} = \frac{y - 1}{- 3} = \frac{z + 2}{1}$ with the plane x + 3y - z + 1 = 0 is

(3, - 1, 1)
(- 5, 1, - 1)
(2, 0, 3)
(4, - 2, - 1)

478.

If the lines $\frac{2 x - 1}{2} = \frac{3 - y}{1} = \frac{z - 1}{3}$ and $\frac{x + 3}{2} = \frac{z + 1}{p} = \frac{y + 2}{5}$ are perpendicular to each other, then p is equal to

1
- 1
10
- $\frac{7}{5}$

479.

The point P(x, y, z) lies in the first octant and its distance from the origin is 12 units. If the position vector of P make 45° and 60° with the x-axis and y-axis respectively, then the coordinates of P are