If x-a cos A-b sin A and y-b cos A-a sin A- prove that x2-y2-a2-b2-

X2 + y2 = (aCos A bSinA)2 + (bCosA + aSinA)2 = a2Cos2A + b2Sin2A 2aCosAbSinA + b2Cos2A + a2Sin2A+2aSinbCos = a2(Cos2 A+ Sin2 A) + b2 ( Cos2 A+ Sin2A) + 2ab ...

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Standard XII

Mathematics

Implicit Differentiation

If x=a cos A-...

Question

If x = a cos A - b sin A and y = b cos A + a sin A, prove that x² + y² = a² + b² .

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Solution

x²+ y² = (aCos A- bSinA)² + (bCosA + aSinA)²
= a²Cos²A + b²Sin²A - 2aCosAbSinA + b²Cos²A + a²Sin²A+2aSinbCos
= a²(Cos² A+ Sin² A) + b² ( Cos² A+ Sin²A) + 2ab ( SinACosA - SinACosA)

Cos² A+ Sin² A = 1
SinACosA - SinACos A= 0

= (a² x 1) + (b² x 1) + 0
= a² + b²

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If x = a cos A - b sin A and y = b cos A + a sin A, prove that x2 + y2 = a2 + b2 .

x2 + y2 = (aCos A- bSinA)2 + (bCosA + aSinA)2 = a2Cos2A + b2Sin2A - 2aCosAbSinA + b2Cos2A + a2Sin2A+2aSinbCos = a2(Cos2 A+ Sin2 A) + b2 ( Cos2 A+ Sin2A) + 2ab ( SinACosA - SinACosA) Cos2 A+ Sin2 A = 1 SinACosA - SinACos A= 0 = (a2 x 1) + (b2 x 1) + 0 = a2 + b2

If x = a cos A - b sin A and y = b cos A + a sin A, prove that x² + y² = a² + b² .

x²+ y² = (aCos A- bSinA)² + (bCosA + aSinA)²
= a²Cos²A + b²Sin²A - 2aCosAbSinA + b²Cos²A + a²Sin²A+2aSinbCos
= a²(Cos² A+ Sin² A) + b² ( Cos² A+ Sin²A) + 2ab ( SinACosA - SinACosA)

Cos² A+ Sin² A = 1
SinACosA - SinACos A= 0

= (a² x 1) + (b² x 1) + 0
= a² + b²