Consider the planes P1: x=y sin α+z sin β P2: y=z sin θ+x sin α P3: z=x sin β+y sin θ If mathrmP1, mathrmP2, mathrmP3 intersect in a line ( mathrmx/ cos θ)=( mathrmy/ cos β)=( mathrmz/ cos α) then

Question

Consider the planes P1: x=y sin α+z sin β P2: y=z sin θ+x sin α P3: z=x sin β+y sin θ If mathrmP1, mathrmP2, mathrmP3 intersect in a line ( mathrmx/ cos θ)=( mathrmy/ cos β)=( mathrmz/ cos α) then (A)  sin 2 α+ sin 2 β+2 sin θ sin α sin β= cos 2 θ (B)  sin 3 α+ sin 3 β+ sin 3 θ=3 sin α sin β sin θ (C) θ+α+β=(π/2) (D) θ+α+β=π

Tardigrade Admin · Accepted Answer

Three planes intersect in a line ∴ mathrmD= mathrmD1= mathrmD2= mathrmD3=0 Here mathrmD1= mathrmD2= mathrmD3=0, ∴ mathrmD=0 Δ=0 | beginarrayccc-1 sin α sin β sin α -1 sin θ sin β sin θ -1 endarray|=0 -1(1- sin 2 θ)- sin α(- sin α- sin θ sin β) + sin β( sin α sin θ+ sin β)=0 sin 2 θ+ sin 2 α+ sin 2 β+2 sin α sin β sin θ=1 Now D.R's of line | beginarrayccc mathrmi mathrmj mathrmk -1 sin α sin β sin α -1 sin θ endarray| =i( sin α sin θ+ sin β)+j( sin α sin β+ sin θ)+k(1- sin 2 α) This vector is parallel to line. ∴ ( sin α sin θ+ sin β/ cos θ)=( sin α sin β+ sin θ/ cos β)=(1- sin 2 θ/ cos α) sin α sin θ+ sin β= cos θ cos α cos (θ+α)= cos ((π/2)-β) ∴ α+β+θ=2 mathrmn π+(π/2)

Q. Consider the planes
$P_{1} : x = y sin α + z sin β$
$P_{2} : y = z sin θ + x sin α$
$P_{3} : z = x sin β + y sin θ$
If $P_{1}, P_{2}, P_{3}$ intersect in a line $\frac{x}{c o s θ} = \frac{y}{c o s β} = \frac{z}{c o s α}$ then

$sin^{2} α + sin^{2} β + 2 sin θ sin α sin β = cos^{2} θ$

$sin^{3} α + sin^{3} β + sin^{3} θ = 3 sin α sin β sin θ$

$θ + α + β = \frac{π}{2}$

$θ + α + β = π$

Q. Consider the planes P1​:x=ysinα+zsinβ P2​:y=zsinθ+xsinα P3​:z=xsinβ+ysinθ If P1​,P2​,P3​ intersect in a line cosθx​=cosβy​=cosαz​ then

sin2α+sin2β+2sinθsinαsinβ=cos2θ

sin3α+sin3β+sin3θ=3sinαsinβsinθ

θ+α+β=2π​

θ+α+β=π

Q. Consider the planes
$P_{1} : x = y sin α + z sin β$
$P_{2} : y = z sin θ + x sin α$
$P_{3} : z = x sin β + y sin θ$
If $P_{1}, P_{2}, P_{3}$ intersect in a line $\frac{x}{c o s θ} = \frac{y}{c o s β} = \frac{z}{c o s α}$ then

$sin^{2} α + sin^{2} β + 2 sin θ sin α sin β = cos^{2} θ$

$sin^{3} α + sin^{3} β + sin^{3} θ = 3 sin α sin β sin θ$

$θ + α + β = \frac{π}{2}$

$θ + α + β = π$