In 3-D space, let three lines L1, L2 and L3 be such that L1 : intersecting the z-axis at P(0, 0, 2) and does not meet the x-y plane L2 : passing through the origin and through the point P. L3 : passing through the origin and making positive angles (α, β, γ) with co-ordinate axes and 45° angle with line L1 Identify the which of the following statement(s) is(are) correct?

Engineering

Mathematics

Equation of Straight Line in 3D

Question

In 3-D space, let three lines L₁, L₂ and L₃ be such that
L₁ : intersecting the z-axis at P(0, 0, 2) and does not meet the x-y plane
L₂ : passing through the origin and through the point P.
L₃ : passing through the origin and making positive angles (α, β, γ) with co-ordinate axes and 45° angle with line L₁
Identify the which of the following statement(s) is(are) correct?

area of the triangle formed by the lines L₁, L₂ and L₃ is 8 square units.

If β = 60°, then equation of L₁ is x + y = 0 and z = 2.

If β = 60°, then equation of L₁ is x = y; z = 2.

area of the triangle formed by the lines L₁ , L₂ and L₃ is 2 square units.

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L₁ : $\frac{x}{a} = \frac{y}{b}$ ; z = 2 (parallel to xy-plane)
L₂ : z-axis
L₃ : $\frac{x}{ℓ} = \frac{y}{m} = \frac{z}{n}$
L₃ is making 45° with z-axis
$∴ ∆$ formed by L₁, L₂, L₃ is isosceles right angled
$∴$ Area of the $∆$ is = $\frac{1}{2}$ × 2 × 2 = 2 square units
Now, β = 60° $\Rightarrow m = \frac{1}{2}, n = \frac{1}{\sqrt{2}} \Rightarrow ℓ = \frac{1}{2}$
$∴$ dr's of L₃ are 1, 1, $\sqrt{2}$

Applying, angle between L₁ and L₃
cos 45° = $\frac{a + b + 0}{\sqrt{a^{2} + b^{2}} \sqrt{1 + 1 + 2}}$
$\frac{1}{\sqrt{2}} = \frac{a + b}{2 \sqrt{a^{2} + b^{2}}} \Rightarrow$ 2(a² + b²) = (a + b)² ⇒ (a – b)² = 0 ⇒ a = b
$∴$ Equation of L₁ is : x = y; z = 2