Sketch the graph of 4x^2 + 9y^2 = 144. Is this an ellipse?

In summary, the formula for an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center, and a and b are the lengths of the semi-major and semi-minor axes. An equation represents an ellipse if the coefficients of x^2 and y^2 are both positive and the equation is in the standard form of an ellipse. To sketch the graph of an ellipse, find the center and lengths of the semi-major and semi-minor axes, plot the center point, and draw the ellipse using the lengths as endpoints. An equation is an ellipse if it is in the standard form and the coefficients of x^2 and y^2 are
  • #1
kasse
384
1
I'm going to sketch the graph of the eq. 4x^2 + 9y^2 = 144

This is an ellipse with its center at the origo and major semiaxis 6 and minor semiaxis 4. But how do I find the foci?
 
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  • #2
[tex]a^2=b^2+c^2[/tex]
where c is the focal length.

the equation comes directly from one definition of ellipse:
the sum of distance between any point on the ellipse and the foci = 2a.
 
  • #3
Ah, makes sense. Thank you!
 

1. What is the formula for an ellipse?

The general formula for an ellipse is (x-h)^2/a^2 + (y-k)^2/b^2 = 1, where (h,k) is the center of the ellipse and a and b are the lengths of the semi-major and semi-minor axes, respectively.

2. How do you determine if an equation represents an ellipse?

An equation represents an ellipse if the coefficients of x^2 and y^2 are both positive and the equation is in the standard form of an ellipse.

3. How do you sketch the graph of an ellipse?

To sketch the graph of an ellipse, first find the center and the lengths of the semi-major and semi-minor axes. Then, plot the center point and draw a horizontal and vertical line through it. From the center, mark off the lengths of the semi-major and semi-minor axes on each line and draw the ellipse using those points as the endpoints of the major and minor axes.

4. How can you tell if the given equation is an ellipse?

If the given equation is in the standard form of an ellipse and the coefficients of x^2 and y^2 are both positive, then it is an ellipse. If the coefficients are not both positive, it may be a different type of conic section such as a circle, parabola, or hyperbola.

5. What are the properties of an ellipse?

An ellipse has two foci, which are points inside the ellipse that determine the shape and size of the ellipse. The distance from the center to each focus is equal to the length of the semi-major axis. The sum of the distances from any point on the ellipse to the two foci is constant. It also has two vertices, which are points on the edge of the ellipse that are farthest from each other. The length of the semi-major axis is equal to the distance between the two vertices.

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