Cubic and Ellipse problem

[QuantumKing]

The question I am looking at states:
Prove that the elliptical function x^2 + 3y^2=b is orthogonal to the cubic y=3ax^3.

I'm thinking they want me to prove if the functions are orthogonal when they intersect. If I am correct that would just lead to;

For the ellipse,
2x+6y(m1)=0, m1=-x/3y, when they intersect: m1=-x/3(3ax^3)=-1/9ax^2

Cubic,
m2=9ax^2

m1xm2=[-1/9ax^2][9ax^2]=-1, Therefore they are orthogonal when they intersect.

Am I right here? Is this even what the question was asking for?

 

[HallsofIvy]

The question I am looking at states:
Prove that the elliptical function x^2 + 3y^2=b is orthogonal to the cubic y=3ax^3.

I'm thinking they want me to prove if the functions are orthogonal when they intersect. If I am correct that would just lead to;

For the ellipse,
2x+6y(m1)=0, m1=-x/3y, when they intersect: m1=-x/3(3ax^3)=-1/9ax^2

Cubic,
m2=9ax^2

m1xm2=[-1/9ax^2][9ax^2]=-1, Therefore they are orthogonal when they intersect.

Am I right here? Is this even what the question was asking for?

Yes, that's exactly right.

 

[tacman]

Yes, you are correct in your approach to the problem. The question is asking for a proof that the two functions are orthogonal at their points of intersection. Your calculation of the slopes at the points of intersection shows that the product of the slopes is -1, which is the condition for two lines to be orthogonal. This proves that the two functions are orthogonal at their points of intersection. Good job!