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## Precalculus (2018 edition)

### Course: Precalculus (2018 edition)>Unit 2

Lesson 11: Identifying conic sections from their expanded equations

# Conic section from expanded equation: ellipse

Sal manipulates the equation 9x^2+4y^2+54x-8y+49=0 in order to find that it represents an ellipse. Created by Sal Khan.

## Want to join the conversation?

• Where can i find the videos about equations of elipses, circles and parabolas? I'm afraid i don't really understand those completely.
• see introduction to conic sections to get started...
(1 vote)
• Is it just me, or do you ever stop after finishing an equation as long and complicated as this and just feel really intelligent? I know in the long run this equation isn't super hard, but the feeling of completing something that crazy-looking is super empowering.
• I understand that an equation for an ellipse is always set equal to one, not zero. Why is this?
• You could subtract 1 from both sides of the equation, and then it would be set equal to zero.
It is just one of several conventions for the equations of circles, ellipses, and hyperbolae to be presented in this form, whereas the equations of parabolae tend to be presented in the form ax² + bx + c = 0.
However, the general form for the equation of any conic section is:
Ax² + By² + Cxy + Dx + Ey + F = 0
Therefore, depending on context, you may see different conventions followed.
For an ellipse, there does need to be a non-zero constant term in order to obtain a curve of any kind (as opposed to a single point). Whether you place the constant on the left or right side of the equation is a matter of taste, but the x² and y² terms are both necessarily positive, so a constant on the oposite side of the equation must be positive, while if placed on the same side it would have to be negative, otherwise there would be no real solutions to the equation, and therefore no points on the coordinate plane could be plotted and there would be no ellipse.
(1 vote)
• How do I know if an equation is representing an ellipse but not a circle?
• An easy way to tell the difference between an ellipse and a circle is if their radii are the same when the equation is in standard form (the way it was after Sal completed the square). For example, if the number under the (x-h)^2 and the number under the (y-k)^2 are equal, then you have a circle. If they are not equal, the radii are different lengths, so the equation is an ellipse.
• How exactly, though, does Sal know that it is an ellipse? By just looking at it? I think I need a bit of clarification in terms of knowing what figures are which by just looking at the equation. Thanks.
• In the conic sections -
Hyperbola - the 2nd degree variables(x^2 and y^2) need to have opposite signs
Parabola - needs to have only 1 of the variables(x or y) as square while the other is degree 1(just x or y).
Ellipse - needs to have both variables in degree 2
Circle - special ellipse
Looking at the above terms we can easily rule out Parabola and Hyperbola. So Sal says that it is probably an ellipse.
• what about in the last equation - it equals 1 but if you have x and y -3 and 1 doesn't it have to be 0?
(1 vote)
• That's the center. It equals 1 when it is on the conic section but the center is never on the shape
• at , where did you get the 81 to add to the right hand side?
(1 vote)
• I'm not sure if something is wrong with just my computer or something but it seems like this video is exactly the same as the "Conic Section from Expanded Equation: Circle and Parabola" I think there might've been some kind of error in the upload?
(1 vote)
• Refresh. It often happens to me. Refreshing again and again will fix it.
• What is standard form?
• standard form is the default form for the type of function you want to graph. I don't know why but I have answered the what.
(1 vote)
• x^2-4xy+3x+25y-6=0
I don't understand how to put this in standard form.
What type of conic is this?
(1 vote)
• x²-4xy+3x+25y-6=0
-4xy+25y+3x+x²-6=0
y(-4x+25)= -x²-3x+6
y(4x-25)=x²+3x-6
y=(x²+3x-6)/(4x-25), for x≠25/4

This is not a conic section, just a rational function. It has a vertical asymptote at x=25/4, goes to +∞ to the right and -∞ to the left.