Answer
$y=2x^2$
Work Step by Step
In order to find the equation of the parabola we need to find the curvature ,for this we will have to use formula 11, such that
$\kappa(x)=\dfrac{|f''(x)|}{[1+(f'(x))^2]^{3/2}}$
Consider $f(x)=ax^2+bx$ [ General equation of a parabola]
$f'(x)=2ax+b$ and $f''(x)=2a$
$\kappa(x)=\dfrac{|2a|}{[1+(2ax+b)^2]^{3/2}}$
As we are given that the $ \kappa(0)$ is the curvature at origin.
Therefore,
$\kappa(0)=\dfrac{|2a|}{[1+b^2]^{3/2}}$
$\implies 4=\dfrac{|2a|}{[1+b^2]^{3/2}}$
or, $a=\pm 2 (1+b^2)^{3/2}$
Hence, the equation of parabola will be: $y=\pm 2 (1+b^2)^{3/2}x^2+bx$
If we take the parabola to have the vertex at the origin, then $b=0$ and $y=2x^2$
