If a line makes an angles - - - - - with positive axes- Then the range of sin-sin- -sin-sin- -sin-sin- is-

We know, sin^2α +sin^2β +sin^2γ =2 nbsp;and (sinα +sinβ +sinγ )^2≥ 0 ⇒ 2+2 (Σsinαsinβ )≥ 0 ⇒Σsinαsinβ≥ 1.....(1) Also, Σ (sinα sinβ )^2≥ 0 ⇒2(sin^2α +sin^ ...

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Standard XII

Mathematics

Position of a Point with Respect to Circle

If a line mak...

Question

If a line makes an angles $α, β, γ$ with positive axes. Then the range of $sin α sin β + sin β sin γ + sin α sin γ$ is:

[\frac{- 1}{2}, 1]

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[\frac{1}{2}, 2]

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[- 1, 2]

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(- 1, 2]

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Solution

The correct option is C $[- 1, 2]$
We know, ${sin}^{2} α + {sin}^{2} β + {sin}^{2} γ = 2$
and $(sin α + sin β + sin γ)^{2} \geq 0$
$\Rightarrow 2 + 2 (Σ sin α sin β) \geq 0$
$\Rightarrow Σ sin α sin β \geq - 1..... (1)$
Also, $Σ (sin α - sin β)^{2} \geq 0 \Rightarrow 2 ({sin}^{2} α + {sin}^{2} β + {sin}^{2} γ) - 2 Σ sin α sin β \geq 0 \Rightarrow 4 - 2 Σ sin α sin β \geq 0$
$\Rightarrow 2 Σ sin α sin β \leq 4 \Rightarrow Σ sin α sin β \leq 2..... (2)$
Range $= [- 1, 2]$

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Position of a Point with Respect to Circle

Standard XII Mathematics

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If a line makes an angles α,β,γ with positive axes. Then the range of sinαsinβ+sinβsinγ+sinαsinγ is:

If a line makes an angles $α, β, γ$ with positive axes. Then the range of $sin α sin β + sin β sin γ + sin α sin γ$ is: