The order of the differential equation of all tangent lines to the parabola y=

Question -

The order of the differential equation of all tangent lines to the parabola $y = x^{2}$ is

Options:

A . 1

B . 2

C . 3

D . 4

Answer: Option A
:
A
The parametric form of the given equation is

x = t, y = t^{2}

.The equation of any tangent at t is

2 x t = y + t^{2}

, Differentiating we get

2 t = y_{1} (= \frac{d y}{d x})

putting this value in the above equation, we have

2 x \frac{y_{1}}{2} = y + {(\frac{y_{1}}{2})}^{2} \Rightarrow 4 x y_{1} = 4 y + y_{1}^{2}

The order of this equation is 1
Hence (A) is the correct answer

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