Classifying a Conic Section (in Standard Form)

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

Related Topics

• Standard Form of a Circle
• How to Write the Equation of Parabola
• Equation of Each Ellipse and Finding the Foci, Vertices, and Co– Vertices of Ellipses
• Hyperbola in Standard Form and Vertices, Co– Vertices, Foci, and Asymptotes of a Hyperbola

Step by Step Guide to Classifying a Conic Section

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

• 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.

• \(\color{blue}{x^2+y^2+6y=7}\)
• \(\color{blue}{x^2-y^2+2x+10y=124}\)
• \(\color{blue}{x^2+2x-4y=-25}\)

• 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: \(\color{blue}{(x-(-1))^2=4(y-6)}\)