Question

if x=asint-bcost, y=acost+bsint, show that D2Y/DX2=-X2+Y2/Y3

Answer 1

x = asint - bcost
y = acost +bsint

x² + y² ={ (asint - bcost)² + ( acost + bsint )²}

x² + y² = a²( sin²t + cos²t ) + b²( sin²t + cos²t )

x² + y² = a² + b²

now , differentiate wrt x

2x + 2y.dy/dx = 0

x + ydy/dx = 0 => dy/dx = -x/y --------(1)

again differentiate , wrt x

1 + yd²y/dx² + (dy/dx)² = 0

put equation (1)

1 + yd²y/dx² + ( -x/y)² = 0

1 + x²/y² + yd²y/dx² = 0

( y² + x² )/y³ = -yd²y/dx²

d²y/dx² = -( x² + y²)/y³
hence proved ///