If you're seeing this message, it means we're having trouble loading external resources on our website.

If you're behind a web filter, please make sure that the domains *.kastatic.org and *.kasandbox.org are unblocked.

### Course: Algebra (all content)>Unit 17

Lesson 11: Identifying conic sections from their expanded equations

# Conic section from expanded equation: circle & parabola

Sal manipulates the equation x^2+y^2-3x+4y=4 in order to find that it represents a circle, and the equation 2x^2+y+12x+16=0 in order to find it represents a parabola. Created by Sal Khan.

## Want to join the conversation?

• So, an ellipse can be a circle, but a circle cannot be an ellipse? Please correct me if I'm wrong.
• I believe in an older video Sal said a circle was a special form of an ellipse
• what is the eccentricity of acircle and why?
• the answer is 0. The eccentricity of an ellipse is dependent on the distance between the foci. As the foci become further appart the eccentricity approaches 1. As the distance between the foci become closer together the eccentricity gets smaller and smaller. When the distance between the foci is zero (a single point) the ellipse is a circle (zero eccentricity).
• y=x^2 is a faunction but x=y^2 is not,right?
• You are correct. The reason for the difference is that y=x² gives one y value for every x value, that makes it a function. x=y² gives two y values when x >0, so it is not a function.
• how would a coefficient effect the shape of a circle, ellipse, or hyperbola?
• It affects where the shape of the conic section will be placed on the graph.
(1 vote)
• How can you identify circles or eclipses whith out solving the problem?
• If the x^2 and y^2 terms have the same coefficient, then it is a circle. If the terms have different coefficients, then it is an ellipse.
Ex: x^2 + y^2 = 9 is a circle where as x^2 + 3y^2 = 9 is an ellipse.
• At how am i supposed to figure out the radius of the ellipse?
• An ellipse doesn't have a radius, but you can find their major/minor axis, is basically the longest (major) and the shortest (minor) distance from the center to a point.

Given the equation: x^2/3^2 + y^2/2^2
You can decide that the major axis is horizontal and the minor axis is vertical. This is determined by finding the larger number in the denominator, which is the major axis. The smaller number is the minor axis.

Now finding the lengths of the major and minor axes:
First, you take the square root of the number in the denominator. (For major, this is 3; for minor, 2) This is only half of the entire length since this is only the distance from the center to a certain point, so we multiply these numbers by 2.
So the answers: Major axis = 6; Minor axis = 4
• I need help with the order pairs and plotting the points
• how do you remember ellipse, circle, parabola, and hyperbola and what they mean?
(1 vote)
• These are simply vocabulary terms. They just need to be memorized. Their formulas, however, can be proven with basic knowledge of their defintion, as Sal demonstrates in the videos in the Conics topic following this one.