Classifying a Conic Section (in Standard Form)

To classify a conical section, remember the standard form of each.

Classifying a Conic Section (in Standard Form)

Conic Section can be represented by a cross-section of a plane cutting through a cone.

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Step by Step Guide to Classifying a Conic Section

The formula for four basic conic sections are provided in table below:

Conic sectionStandard form of equation
Parabola  \((x- h)^2= 4p(y-k)\), \((y+k)^2= 4p(x-h)\)
Circle \((x- h)^2+( y-k)^2=r^2\)
Ellipse \(\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1\), \(\frac{(x-h)^2}{b^2}+\frac{(y-k)^2}{a^2}=1\)
Hyperbola  \(\frac{(y-k)^2}{a^2}-\frac{(x-h)^2}{b^2}=1\), \(\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1\)
  • For identify a conic section Group \(x\) and \(y\) variables together, then convert \(x\) and \(y\) to square form.

Classifying a Conic Section – Example 1:

Write this equation in standard form: \(x^2+y^2+12x=-11\)

Solution:

Group \(x\)-variables and \(y\)-variables together: \((x^2+12x+36)+y^2=-11\)

Convert \(x\) to square form: \((x^2+12x+36)+y^2=-11+36\) \((x^2+12x+36)+y^2=25\)

Then: \((x+6)^2+y^2=5^2\), its a circle.

Exercises for Classifying a Conic Section

Write teach equation in standard form.

  • \(\color{blue}{x^2+y^2+6y=7}\)
  • \(\color{blue}{x^2-y^2+2x+10y=124}\)
  • \(\color{blue}{x^2+2x-4y=-25}\)
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  • It’s a circle: \(\color{blue}{x^2+(y+3)^2=16}\)
  • It’s a hyperbola: \(\color{blue}{\frac{(x-(-1))^2}{10^2}-\frac{y-5}{10^2}=1}\)
  • It’s parabola: \(\color{blue}{(x-(-1))^2=4(y-6)}\)

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