Finding the Value of C for Orthogonal Intersection of Two Curves

[anthonym44]

Homework Statement

Two curves intersect orthogonally when their tangent lines at each point of intersection are perpendicular. Suppose C is a positive number. The curves y=Cx^2 and y=(1/x^2) intersect twice. Find C so that the curves intersect orthogonally. For that value of C, sketch both curves when -2 < x < 2 and 0 < x < 4

Homework Equations

y=(1/x^2)
y'=(-2/x^3)

The Attempt at a Solution

The tangent line at the the other equation (y=Cx^2) must be the opposite reciprocal of the tangent line y'=(-2/x^3), therefore it must =(x^3/2) this has to equal the derivative of the equation y=(Cx^2) for this equation to have this derivative C must = (x^2/8) because when subsituted into the equation y=(Cx^2)you get y=(x^2/8)(x^2) which is equal to y=(x^4/8) when you find its derivative you get y'= (x^3/2).
I'm not sure if I am going about doing this correctly if anyone can point me in the right direction please give me anything you have to offer. thanks

 

[EnumaElish]

At which point(s) do they intersect?

 

[anthonym44]

(1.4,.5), (-1.4,.5)

 

[EnumaElish]

(1.4,.5), (-1.4,.5)

How is it that these points do not involve terms with C? (I don't understand how or why they are independent of C.)