For the function f(x,y) = e^( -(5x^2 + 6y^2) ), what are the shape of the level curves?

Question

For the function f(x,y) = e^( -(5x^2 + 6y^2) ), what are the shape of the level curves?

A collection of equally spaced parallel lines,

A collection of unequally spaced parallel lines,

A collection of intersecting lines,

A collection of equally spaced concentric circles,

A collection of unequally spaced concentric circles,

A collection of concentric ellipses,

A collection of parabolas,

One straight line and a collection of parabolas,

Two straight lines and a collection of hyperbolas,

None of the above

Answer 1

Level curves are when the Z value (or function) is constant.

If we set the function equal to a constant we get:

C = e^( -(5x^2 + 6y^2) )

lnC = - (5x^2 + 6y^2)

-lnC = 5x^2 + 6y^2

Varying the value of C means that the level curves will be concentric ellipses since the above is almost the standard form of an ellipse.