Formation of the differential equation corresponding to the ellipse major axis 2a and minor axis 2b is:a)b)c)d)Correct answer is option 'A'. Can you explain this answer?

Question

Answer

Equation of ellipse :

x²/a² + y²/b² = 1

Differentiation by x,

2x/a² + (dy/dx)*(2y/b²) = 0

dy/dx = -(b²/a²)(x/y)

-(b²/a^2) = (dy/dx)*(y/x) ----- eqn 1

Again differentiating by x,

d²y/dx² = -(b²/a²)*((y-x(dy/dx))/y²)

Substituting value of -b²/a² from eqn 1

d²y/dx² = (dy/dx)*(y/x)*((y-x(dy/dx))/y²)

d²y/dx² = (dy/dx)*((y-x*(dy/dx))/xy)

(xy)*(d²y/dx²) = y*(dy/dx) - x*(dy/dx)²

(xy)*(d²y/dx²) + x*(dy/dx)²- y*(dy/dx) = 0

Answer

General equatioon x^2/a^2+y^2/b^2=1 , y^2=b^2-{b^2/a^2}x^2

differentiating both side 2ydy/dx= {-b^2/a^2}2x

[ ydy/dx= {-b^2/a^2}x ]

again diff^n y{d^2y/d^2x} + {dy/dx}^2 = - {b^2/a^2}

multiplied by x both side xy{d^2y/d^2x} +x {dy/dx}^2 = - {b^2/a^2}x

[xy{d^2y/d^2x} +x {dy/dx}^2 +x {b^2/a^2}] Ans

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