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What is the differential equation to family of parabolas having their vertices at the origin and foci on the x-axis?1. y-2xy' = 02. x = 2yy'3. xy = y'4. x = yy'

Answer» Correct Answer - Option 1 : y-2xy' = 0

Concept:

The equation of parabola having their vertices at the origin and foci on the x-axis is \(\rm y^2 = 4ax \)

 

Calculations:

The equation of parabola having their vertices at the origin and foci on the x-axis is

 \(\rm y^2 = 4ax \)              ....(1)

Differentiating on both side, we get

\(\rm 2y \dfrac {dy}{dx} = 4a\)

⇒ \(\rm y \dfrac {dy}{dx} = 2a\)

⇒ \(\rm a = \dfrac y2 \dfrac {dy}{dx} \)

Put this value in equation (1), we get

\(\rm y^2 = 4\dfrac y2 \dfrac {dy}{dx} x \)

⇒ \(\rm y^2 - 2xy\dfrac {dy}{dx} = 0\)

⇒ y(y - 2xy') = 0, Here y' = \(\rm \frac{dy}{dx}\)

∴ y-2xy' = 0

Hence,  the differential equation to the family of parabolas having their vertices at the origin and foci on the x-axis is y - 2xy' = 0


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