Question

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

• A. $1$
• B. $2$
• C. $3$
• D. $4$

Answer 1

The correct option is A

$1$

Evaluate the order of the differential equation

The order of a differential equation is defined as the highest order of derivative of the differential equation.

The degree of the differential equation is defined as the power of the highest order of derivative of the differential equation.

The parametric form of the given equation is $x=t,y=t^2$

The equation of any tangent at $t$ is

$2xt=y+t^2$ ……….$\left(i\right)$

On differentiating, we get

$2t=y_1$ ……..$\left(ii\right)$

Now, using $\left(ii\right)$ in $\left(i\right)$ we have

$$\begin{gathered} x{y}_1=y+{\left(\frac{y_1}{2}\right)}^2 \\ \Rightarrow 4x{y}_1=4y+{\left(y_1\right)}^2 \end{gathered}$$

So the order of the differential equation is $1$

Hence, the correct option is $\left(\mathrm{A}\right)$.