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.

Q: The equation of the straight line through the origin parallel to the line
(b + c)x + (c + a)y + (a + b)z = k = (b – c)x + (c – a)y + (a – b)z is

(A) $\large \frac{x}{b^2-c^2} = \frac{y}{c^2-a^2} = \frac{z}{a^2-b^2} $

(B) $\large \frac{x}{a} = \frac{y}{b} = \frac{z}{c} $

(C) $\large \frac{x}{a^2-bc} = \frac{y}{b^2-ca} = \frac{z}{c^2-ab} $

(D) none of these

Sol. Equation of straight line through origin is

$\large \frac{x-0}{l} = \frac{y-0}{m} = \frac{z-0}{n} $

where l((b+c) + m(c+a) + n(a + b) = 0

and l(b – c) + m(c -a) + n(a – b) = 0

On solving,

$\large \frac{l}{2(a^2-bc)} = \frac{m}{2(b^2-ca)} = \frac{n}{2(c^2-ab)} $

Equation of the straight line is

$\large \frac{x}{a^2-bc} = \frac{y}{b^2-ca} = \frac{z}{c^2-ab} $

Hence (C) is the correct answer.

Q: A straight line passes through the point (2, –1, –1). It is parallel to the plane  4x + y + z + 2 = 0 and is perpendicular to the line $ \frac{x}{1} = \frac{y}{-2} = \frac{z-5}{1} $ . The equation of the straight line is

(A) $\large \frac{x-2}{4} = \frac{y+1}{1} = \frac{z+1}{1} $

(B) $\large \frac{x+2}{4} = \frac{y-1}{1} = \frac{z-1}{1} $

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

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

Sol. Let direction cosines of straight line be l, m, n

∴ 4l + m + n = 0

l – 2m + n = 0

$ \frac{l}{3} = \frac{m}{-3} = \frac{n}{-9} $

$ \frac{l}{-1} = \frac{m}{1} = \frac{n}{3} $

Equation of straight line is

$\large \frac{x-2}{-1} = \frac{y+1}{1} = \frac{z+1}{3} $

Hence (C) is the correct answer.

Q: The shortest distance between the two straight lines $\frac{x-4/3}{2} = \frac{y + 6/5}{3} = \frac{z-3/2}{4} $ and $\frac{5y+6}{8} = \frac{2z-3}{9} = \frac{3x-4}{5} $ is

(A) $\sqrt{29}$

(B) 3

(C) 0

(D) $6 \sqrt{10}$

Sol. Since these two lines are intersecting so shortest distance between the lines will be 0.

Hence (C) is the correct answer.

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

(A) 23x – 13y + 32z + 45 = 0

(B) 5x – y – 4z = 3

(C) 5x – y – 4z + 45 = 0

(D) 23x – 13y + 32z + 3 = 0

Sol. a₁a₂ + b₁ b₂ + c₁c₂ = 6 + 2 + 12 =positive

So acute angle bisector is $ \large \frac{2x – y + 2z +3}{3} = – \frac{3x-2y + 6z + 8 }{7} $

⇒ 23x – 13y + 32z + 45 = 0

Hence (A) is the correct answer.

Q: Consider the following statements:
Assertion (A): the plane y + z + 1 = 0 is parallel to x-axis.
Reason (R): normal to the plane is parallel to x-axis.
Of these statements:

(A) both A and R are true and R is the correct explanation of A

(B) both A and R are true and R is not a correct explanation of A

(C) A is true but R is false

(D) A is false but R is true

Sol: Given plane y + z + 1 = 0 is parallel to x-axis as 0.1 + 1.0 + 1.0 = 0

but normal to the plane will be perpendicular to x-axis.

Hence (C) is the correct answer.

Q: A variable plane at a distance of 1 unit from the origin cuts the co-ordinate axes at A, B and C. If the centroid D (x, y, z) of triangle ABC satisfies the relation $\large \frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = k $ , then the value of k is

(A) 3

(B) 1

(C) 1/3

(D) 9

Sol: Let $\large \frac{x}{a} + \frac{y}{a} + \frac{z}{a} = 1 $ be the variable plane so that

$\large \frac{1}{\sqrt{\frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2}}} = 1 $

The plane meets the coordinate axes at A (a, 0, 0), B (0, b, 0), C (0, 0, c). The centroid D of the triangle ABC is $(\frac{a}{3} , \frac{b}{3} , \frac{c}{3})$

Þ x = a/3 , y = b/3 , z = c/3

and $\large \frac{1}{a^2} + \frac{1}{b^2} + \frac{1}{c^2} = 1 $

$\large \frac{1}{x^2} + \frac{1}{y^2} + \frac{1}{z^2} = 9 $

Hence (D) is the correct answer