How do you identity if the equation #4x^2+2y^2=8# is a parabola, circle, ellipse, or hyperbola and how do you graph it?

1 Answer
Dec 16, 2016

Answer:

The reference for Conic Section - General Cartesian form tells you how to determine what conic section it is.

Explanation:

The reference for Conic Section - General Cartesian form tells you how to determine what conic section it is, when given the General Cartesian form:

#Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0#

Here is the given equation in the general form:

#2x^2 + 4y^2 - 8 = 0#

Please observe the value of #B^2 - 4AC = 0^2 - 4(2)(4) = -32#. The reference says that this is an ellipse.

Divide both sides of the original equation by 8:

#x^2/4 + y^2/2 = 1#

Write the denominators as squares:

#x^2/2^2 + y^2/(sqrt(2))^2 = 1#

Insert zeros within the squares in the numerators:

#(x - 0)^2/2^2 + (y - 0)^2/(sqrt(2))^2 = 1#

This is the standard form for an ellipse, because it is easy to see:

  1. The center is (0, 0)
  2. When #y = 0, x = -2 and 2#
  3. When #x = 0, y = -sqrt(2) and sqrt(2)#

You may use these four points to graph the equation:

#(2, 0), (0, sqrt(2)), (-2, 0), and (0,-sqrt(2))#

And then sketch in an ellipse around the center.

Here is a graph of the equation:

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