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 $1$

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

The equation of any tangent at $t$ is $2xt=y+t^2$.

Differentiating, we get $2t=y_1$.

Putting this value in the equation of tangent, we have

$2x\frac{y_1}{2}=y+{\left(\frac{y_1}{2}\right)}^2\Rightarrow 4x{y}_1=4y+y_1^2$
The order of this equation is one.