Question

If ax +by =p and bx-ay =q and a²+b²=1 then prove that p²+q²=x²+y²​

Answer 1

Required Answer:-

Given:

• ax + by = p
• bx - ay = q
• a² + b² = 1

To prove:

• p² + q² = x² + y²

Proof:

Given that,

➡ ax + by = p

➡ bx - ay = q

So,

p² + q²

= (ax + by)² + (bx - ay)²

= (ax)² + (by)² + 2axby + (bx)² + (ay)² - 2bxay

= a²x² + a²y² + b²x² + b²y² + 2abxy - 2abxy

= a²x² + a²y² + b²x² + b²y²

= a²(x² + y)² + b²(x² + y²)

= (a² + b²)(x² + y²)

It's mentioned that a² + b² = 1

So,

(a² + b²)(x² + y²)

= 1 × (x² + y²)

= x² + y²

➡ p² + q² = x² + y² (Hence Proved)

Answer 2

Answer:

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