If sin2- - x2-y2-1-2x- then x must be

The correct option is C 1 As 1≤sinθ≤ 1 ⇒ 0≤sin^ 2 θ≤ 1 #160; ⇒ 0≤ x ^ 2 + y ^ 2 +1 2x ≤ 1 Now from #160; x ^ 2 + y ^ 2 +1 2xy ≥ ...

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Byju's Answer

Standard XII

Mathematics

Properties Derived from Trigonometric Identities

If sin2θ = ...

Question

If ${sin}^{2} θ = \frac{x^{2} + y^{2} + 1}{2 x},$ then $x$ must be

- 3

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1

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none of these

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Solution

The correct option is C $1$
As $- 1 \leq sin θ \leq 1$

$\Rightarrow 0 \leq {sin}^{2} θ \leq 1$

$\Rightarrow 0 \leq \frac{x^{2} + y^{2} + 1}{2 x} \leq 1$
Now from

$\frac{x^{2} + y^{2} + 1}{2 x y} \geq 0 \Rightarrow x^{2} + y^{2} + 1 \geq 0$ ...(1)

And from
$\frac{x^{2} + y^{2} + 1}{2 x} \leq 1 \Rightarrow x^{2} + y^{2} + 1 \leq 2 x \Rightarrow {(x - 1)}^{2} + y^{2} \leq 0$ ...(2)

(1) and (2) only holds when $x = 1$ and $y = 0$

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