Question

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

• A. $1,2$
• B. $2,3$
• C. $2,1$
• D. $1,1$

Answer 1

The correct option is A $1,2$

$letpointatwhichtangentisdrawnis(a,a^2)thenslopeoftangent=\left(\frac{dy}{dx}\right)nowy=x^2\left(\frac{dy}{dx}\right)=2x=2a\therefore e{q}^{n}oftangenty-a^2=2a(x-a)y-a^2=2ax-2a^{2}y=2ax-a^{2}now,weneedtofinditsdifferentialequationdifferentiatebothsides,weget\left(\frac{dy}{dx}\right)=2aora=\left(\right(\frac{1}{2}\left)\right(\frac{dy}{dx}\left)\right)\therefore y=2\times \left(\frac{1}{2}\right)\left(\frac{dy}{dx}\right)-{\left(\right(\frac{1}{2}\left)\right(\frac{dy}{dx}\left)\right)}^{2}clearly,order=1anddegree=2$