Showing that two function are orthogoanl trajectories. Please help!!!

[CharismaticBarber]

The problem is as follows:

Show that the family of ellipses 3x^2+y^2=k (k>0) and the family of cubic curves x=cy^3 are orthogonal trajectories of each other.

Now I know that to show this, just show that m1xm2=-1 and I've tried to find the derivatives of both, as well as finding the point of intersection (I got y=4sqrt(1/6c) but I'm pretty sure that's wrong) but I'm really lost with this problem and it'd be great if someone could show how to prove this with all steps involved so I can reference the process in future problems. Thanks!

 

[Ishuda]

The problem is as follows:

Show that the family of ellipses 3x^2+y^2=k (k>0) and the family of cubic curves x=cy^3 are orthogonal trajectories of each other.

Now I know that to show this, just show that m1xm2=-1 and I've tried to find the derivatives of both, as well as finding the point of intersection (I got y=4sqrt(1/6c) but I'm pretty sure that's wrong) but I'm really lost with this problem and it'd be great if someone could show how to prove this with all steps involved so I can reference the process in future problems. Thanks!

The easiest way to do this, IMO, is first look at the derivative of the first function
6 x + 2 y y' = 0
or
y' = - 3x/y


Now look at the derivative for the second function
1 = 3 c y² y' = 3 (c y³) y' / y = (3 x / y) y'
or
y' = 1/(3x/y)

What is y' of the first function times y' of the second function and what does that mean?