Three Dimensional Geometry

Q: The equation of planes bisecting the angle between the planes 2x – y + 2z + 3 = 0 and 3x – 2y + 6z + 8 = 0 is / are

(A) 5x – y – 4z – 45 = 0

(B) 5x – y – 4z – 3 = 0

(C) 23x – 13y + 32z + 45 = 0

(D) 23x – 13y + 32z + 5 = 0

Sol. Equation of planes bisecting the angle between given planes are

$\large \frac{2x-y+ 2z+3}{\sqrt{4+1+4}} = \pm \frac{3x-2y+6z+8}{\sqrt{9+4+36}} $

⇒ 5x – y – 4z – 3 = 0 and 23x – 13y + 32z + 45 = 0 are required planes.

Hence (B) and (C) are the correct answers.

Q: The equations of the lines of shortest distance between the lines $ \large \frac{x}{2} = \frac{y}{-3} = \frac{z}{1} $ and $\large  \frac{x-2}{3} = \frac{y-1}{-5} = \frac{z+2}{2} $ are

(A) 3(x – 21) = 3y + 92 = 3z – 32

(B) $ \large \frac{x-62/3}{1/3} = \frac{y+31}{1/3} = \frac{z-31}{1/3} $

(C) $ \large \frac{x-21}{1/3} = \frac{y + 92/3}{1/3} = \frac{z-32/3}{1/3} $

(D) $ \large \frac{x-2}{1/3} = \frac{y+3}{1/3} = \frac{z-1}{1/3} $

Sol. Let P(2r₁ , – 3r₁, r₁) and Q(3r₂ + 2, – 5r₂ + 1, 2r₂ – 2) be the points on the given lines so that PQ is the line of shortest distance

d.r.s of PQ 2r₁ – 3r₂ – 2 , – 3r₁ + 5r₂ – 1, r₁ – 2r₂ + 2

Since it is perpendicular to given lines

2(2r₁ – 3r₂ – 2) – 3(3r₁ + 5r₂ – 1) + (r₁ – 2r₂ + 2) = 0

and (2r₁ – 3r₂ – 2) – 5(– 3r₁ + 5r₂ – 1) + 2(r₁ – 2r₂ + 2) = 0

r₁ = 31/3, r₂ = 19/3

P is (62/3 , -31 , 31/3) and Q is (21 , -92/3 , 32/3)

d.r.s of PQ is $ (\frac{1}{3} , \frac{1}{3} , \frac{1}{3})$

Hence (A), (B) and (C) are the correct answers.

Q: The lines  $\large  \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z+1}{0} $ and $\large  \frac{x-4}{2} = \frac{y+0}{0} = \frac{z+1}{3} $

(A) are non coplanar

(B) are coplanar

(C) intersecting at (4, 0, – 1)

(D) intersecting at (1, 1, – 1)

Sol: Let $\large  \frac{x-1}{3} = \frac{y-1}{-1} = \frac{z+1}{0} = \lambda_1 $ and $\large  \frac{x-4}{2} = \frac{y+0}{0} = \frac{z+1}{3} = \lambda_2 $

Then 1 + 3λ₁ = 4 + 2λ₂ …(1)

1 – λ₁ = 0 …(2)

– 1 = 3λ₂ – 1 …(3)

λ₁ = 1 , λ₂ = 0 satisfies (1)

lines are intersecting hence coplanar and point of intersection (4, 0, – 1).

Hence (B) and (C) are the correct answers.

Q: The coordinate of the points on the line $\large  \frac{x+2}{3} = \frac{y+1}{2} = \frac{z-3}{2} $ which are at a distance 3√2 from the point (1, 2, 3)

(A) (– 2, – 1, 3)

(B) (2, 2, 4)

(C) $ ( \frac{56}{17} , \frac{43}{17} , \frac{111}{17} ) $

(D) $ ( \frac{47}{11} , \frac{42}{11} , \frac{56}{11} ) $

Sol. Any point on the line (3λ – 2, 2λ – 1, 2λ + 3), then

(3λ – 3)² + (2λ – 3)² + (2λ)² = 18

λ = 0, 13/17

points are (– 2, – 1, 3) and $ ( \frac{56}{17} , \frac{43}{17} , \frac{111}{17} ) $

Hence (A) and (C) are the correct answers.