Are Cubic and Elliptical Functions Orthogonal at Intersection Points?

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SUMMARY

The elliptical function defined by the equation x2 + 3y2 = b is orthogonal to the cubic function y = 3ax3 at their intersection points. The slopes of the tangent lines at the intersection points are calculated as m1 = -1/9ax2 for the ellipse and m2 = 9ax2 for the cubic. The product of these slopes, m1 * m2, equals -1, confirming that the functions are orthogonal at their intersection points. This conclusion directly addresses the original question posed in the discussion.

PREREQUISITES
  • Understanding of calculus, specifically derivatives and slopes of functions.
  • Familiarity with the equations of elliptical and cubic functions.
  • Knowledge of orthogonality in the context of geometry and functions.
  • Basic algebra skills for manipulating equations and solving for variables.
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  • Study the properties of elliptical functions, particularly their geometric interpretations.
  • Learn about cubic functions and their applications in various mathematical contexts.
  • Explore the concept of orthogonality in higher dimensions and its implications in vector calculus.
  • Investigate the use of derivatives in determining the behavior of functions at intersection points.
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Mathematicians, students studying calculus and geometry, and anyone interested in the properties of functions and their intersections.

QuantumKing
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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?
 
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QuantumKing said:
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.
 

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