Question

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If 2y cosθ = x sinθ and, 2x secθ - y cosecθ = 3. Then find the value of (x² + 4y²)




Thanks ;)
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Answer 1

(1)

2y cos∅ = x sin∅  

=> sin∅ = (2y/x) cos∅

(2)

2x sec∅ - ycosec∅ = 3

=> 2x sec∅ - (y/sin∅) = 3

=> (2x/cos∅) - (y/sin∅) = 3

=> cos∅ = (x/2)

We have sin²∅ + cos²∅ = 1

=> (4y²/x² cos²∅) + (x²/4) = 1

=> 4y² + x² = 4

#BeBrainly

Answer 2

AnswEr :

• If 2y cosθ = x sinθ and, 2x secθ - y cosecθ = 3
• Then find the value of (x² + 4y²)

• From Above Equations cosθ = sinθ, this means ;

» (θ + θ) = 90

» 2θ = 90

» θ = 45°

• Using the Value of θ = 45° in Equations :

⇒ 2y cosθ = x sinθ

⇒ 2y cos45° = x sin45°

• cos45° = sin45° = 1 /√2

⇒ 2y × 1 /√2 = x × 1 /√2

• cancelling 1 /√2 both sides

⇒ 2y = x

⇒ x = 2y —( 1 )

⇒ 2x secθ - y cosecθ = 3

⇒ 2x sec45° - y cosec45° = 3

• sec45° = cosec45° = √2

⇒ ( 2x × √2 ) - ( y × √2 ) = 3

⇒ 2√2x - √2y = 3

⇒ (2√2 × 2y) - √2y = 3 —[ from ( 1 )]

⇒ 4√2y - √2y = 3

⇒ 3√2y = 3

• Dividing Each term by 3

⇒ √2y = 1

⇒ y = 1 /√2

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• Now Let's Head to the Question :

⇒ ( x² + 4y² )

⇒ ( 2y )² + 4 × ( y )² —[ from ( 1 )]

⇒ ( 2 × 1 /√2 )² + 4 × ( 1 /√2 )²

⇒ ( 2 /√2 )² + 4 × ( 1 /√2 )²

⇒ ( 4 /2 ) + ( 4 /2 )

⇒ ( 2 + 2 )

⇒ 4

჻ Value of ( x² + 4y² ) is 4.