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Study Guides > College Algebra

Identifying Nondegenerate Conics in General Form

In previous sections of this chapter, we have focused on the standard form equations for nondegenerate conic sections. In this section, we will shift our focus to the general form equation, which can be used for any conic. The general form is set equal to zero, and the terms and coefficients are given in a particular order, as shown below.

Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0
where A,BA,B, and CC are not all zero. We can use the values of the coefficients to identify which type conic is represented by a given equation. You may notice that the general form equation has an xyxy term that we have not seen in any of the standard form equations. As we will discuss later, the xyxy term rotates the conic whenever  B \text{ }B\text{ } is not equal to zero.
Conic Sections Example
ellipse 4x2+9y2=14{x}^{2}+9{y}^{2}=1
circle 4x2+4y2=14{x}^{2}+4{y}^{2}=1
hyperbola 4x29y2=14{x}^{2}-9{y}^{2}=1
parabola 4x2=9y or 4y2=9x4{x}^{2}=9y\text{ or }4{y}^{2}=9x
one line 4x+9y=14x+9y=1
intersecting lines (x4)(y+4)=0\left(x - 4\right)\left(y+4\right)=0
parallel lines (x4)(x9)=0\left(x - 4\right)\left(x - 9\right)=0
a point 4x2+4y2=04{x}^{2}+4{y}^{2}=0
no graph 4x2+4y2=14{x}^{2}+4{y}^{2}=-1

A General Note: General Form of Conic Sections

A nondegenerate conic section has the general form
Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0
where A,BA,B, and CC are not all zero. The table below summarizes the different conic sections where B=0B=0, and AA and CC are nonzero real numbers. This indicates that the conic has not been rotated.
ellipse Ax2+Cy2+Dx+Ey+F=0, AC and AC>0A{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\text{ }A\ne C\text{ and }AC>0
circle Ax2+Cy2+Dx+Ey+F=0, A=CA{x}^{2}+C{y}^{2}+Dx+Ey+F=0,\text{ }A=C
hyperbola Ax2Cy2+Dx+Ey+F=0 or Ax2+Cy2+Dx+Ey+F=0A{x}^{2}-C{y}^{2}+Dx+Ey+F=0\text{ or }-A{x}^{2}+C{y}^{2}+Dx+Ey+F=0, where AA and CC are positive
parabola Ax2+Dx+Ey+F=0 or Cy2+Dx+Ey+F=0A{x}^{2}+Dx+Ey+F=0\text{ or }C{y}^{2}+Dx+Ey+F=0

How To: Given the equation of a conic, identify the type of conic.

  1. Rewrite the equation in the general form, Ax2+Bxy+Cy2+Dx+Ey+F=0A{x}^{2}+Bxy+C{y}^{2}+Dx+Ey+F=0.
  2. Identify the values of AA and CC from the general form.
    1. If AA and CC are nonzero, have the same sign, and are not equal to each other, then the graph is an ellipse.
    2. If AA and CC are equal and nonzero and have the same sign, then the graph is a circle.
    3. If AA and CC are nonzero and have opposite signs, then the graph is a hyperbola.
    4. If either AA or CC is zero, then the graph is a parabola.

Example 1: Identifying a Conic from Its General Form

Identify the graph of each of the following nondegenerate conic sections.
  1. 4x29y2+36x+36y125=04{x}^{2}-9{y}^{2}+36x+36y - 125=0
  2. 9y2+16x+36y10=09{y}^{2}+16x+36y - 10=0
  3. 3x2+3y22x6y4=03{x}^{2}+3{y}^{2}-2x - 6y - 4=0
  4. 25x24y2+100x+16y+20=0-25{x}^{2}-4{y}^{2}+100x+16y+20=0

Solution

  1. Rewriting the general form, we have A=4A=4 and C=9C=-9, so we observe that AA and CC have opposite signs. The graph of this equation is a hyperbola.
  2. Rewriting the general form, we have A=0A=0 and C=9C=9. We can determine that the equation is a parabola, since AA is zero.
  3. Rewriting the general form, we have A=3A=3 and C=3C=3. Because A=CA=C, the graph of this equation is a circle.
  4. Rewriting the general form, we have A=25A=-25 and C=4C=-4. Because AC>0AC>0 and ACA\ne C, the graph of this equation is an ellipse.

Try It 1

Identify the graph of each of the following nondegenerate conic sections.
  1. 16y2x2+x4y9=016{y}^{2}-{x}^{2}+x - 4y - 9=0
  2. 16x2+4y2+16x+49y81=016{x}^{2}+4{y}^{2}+16x+49y - 81=0
Solution

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