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

Answer: (A)

The parametric form of the given equation is $x=t$, $y=t^2$.
The equation of any tangent at $t$ is $2xt=y+t^2$
On differentiating, we get $2t=y_1$
On putting this value in the above equation, we get
$x{y}_1=y+{\left(\frac{y_1}{2}\right)}^2$
$\Rightarrow 4x{y}_1=4y+y_1^2$
The order of this equation is $1$.