Answer:
[tex]y=5x^{2}[/tex]
Step-by-step explanation:
The equation of a parabola is:
[tex]y=ax^{2}+bx+c[/tex]
We know that the vertex is at the vertex, so b and c are 0.
Then, our quadratic equation is rewritten as:
[tex]y=ax^{2}[/tex] (1)
Now, the equation of the curvature related to a function is given by:
[tex]\kappa=\frac{|y''|}{(1+y'^{2})^{3/2}}[/tex] (2)
Here:
y'' is the second derivative of the function
y' is the first equation of the function
κ is the curvature
Let's find the first and the second derivative of y whit respect to x.
[tex]y'=2ax[/tex]
[tex]y''=2a[/tex]
Putting these values in equation (2), we have:
[tex]\kappa (x)=\frac{|2a|}{(1+(2ax)^{2})^{3/2}}[/tex]
We can evaluate the curvature at the origin to find the parameter a. Which means x=0. (Let's recall that κ = 10 at the origin)
[tex]\kappa (0)=|2a|[/tex]
[tex]10=|2a|[/tex]
[tex]|a|=5[/tex]
Now, we have two options for the equation:
[tex]y=5x^{2}[/tex] or [tex]y=-5x^{2}[/tex]
But we know that this parabola opens upward, so the value of a must be positive.
Therefore, the equation is:
[tex]y=5x^{2}[/tex]
I hope it helps you!
