- #1
aisha
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When in standard form how do you know whether the ellipse is horizontal or vertical? and how do you know what a= and b= for the major and minor axis, and vertices? How do you get the coordinates of the foci?
Understanding Ellipse Standard Form and Axis Orientation
In summary, when in standard form, you can determine the horizontal or vertical orientation of an ellipse by looking at the size of the denominators and comparing them. You can also find the foci of an ellipse using the coordinates of the covertex.
- #2
mtong
- 33
- 0
If the denominator attached to the x is less than that of the y then it will be vertical, (as there is less of a distance between the x intercepts that the y).
The semi-major axis will be the root of the largests denominator
The semi-minor axis will be the root of the smallest denominator
The coordinates of the center will be (h,k) for and elispse in the form (just a simple translation)
[tex]\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1[/tex]
The semi-major axis will be the root of the largests denominator
The semi-minor axis will be the root of the smallest denominator
The coordinates of the center will be (h,k) for and elispse in the form (just a simple translation)
[tex]\frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2}=1[/tex]
- #3
aisha
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- 0
how about the coordinates of the foci?
- #4
BobG
Science Advisor
Homework Helper
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If you know what's special about the covertex, there's a very easy way to find the coordinates of each focus.
Both focii lie on the major axis. For a horizontal ellipse, your coordinates have to be (x,0). For a vertical ellipse, (0,y).
The covertex is the point where the minor axis intersects the ellipse. Half of the minor axis is the semi-minor axis (b) and half of the major axis is the semi-major axis (a). The distance from either focus to the covertex is equal to the semi-major axis.
Now you know two sides of your triangle. You know the hypotenuse which happens to have a length equal to the semi-major axis (a). You know the semi-minor axis (b) which forms one leg of your triangle. You also know the minor axis is perpendicular to the major axis, so you know you have a right triangle.
You use the Pythagorean thereom to find the missing length: the length from your origin to the focus. The length will be equal to your linear eccentricity (c). In other words, you have [tex]a^2=b^2+c^2[/tex]. For a horizontal ellipse, the length is the missing x variable in your coordinates. For a vertical ellipse, the missing y variable. For the opposite focii, just reverse the sign (positive x to negative x, pos y to neg y, as applicable).
Both focii lie on the major axis. For a horizontal ellipse, your coordinates have to be (x,0). For a vertical ellipse, (0,y).
The covertex is the point where the minor axis intersects the ellipse. Half of the minor axis is the semi-minor axis (b) and half of the major axis is the semi-major axis (a). The distance from either focus to the covertex is equal to the semi-major axis.
Now you know two sides of your triangle. You know the hypotenuse which happens to have a length equal to the semi-major axis (a). You know the semi-minor axis (b) which forms one leg of your triangle. You also know the minor axis is perpendicular to the major axis, so you know you have a right triangle.
You use the Pythagorean thereom to find the missing length: the length from your origin to the focus. The length will be equal to your linear eccentricity (c). In other words, you have [tex]a^2=b^2+c^2[/tex]. For a horizontal ellipse, the length is the missing x variable in your coordinates. For a vertical ellipse, the missing y variable. For the opposite focii, just reverse the sign (positive x to negative x, pos y to neg y, as applicable).
- #5
aisha
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lol woh bob u got way too complicated how did triangles get into this :rofl:
- #6
BobG
Science Advisor
Homework Helper
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- #7
aisha
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- 0
how do I open the attachment?
1. What is the standard form of an ellipse?
The standard form of an ellipse is (x - h)2/a2 + (y - k)2/b2 = 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 is the standard form of an ellipse different from the general form?
The standard form of an ellipse is a simplified version of the general form, which is Ax2 + Bxy + Cy2 + Dx + Ey + F = 0. In the standard form, the coefficients A and C are equal and B = 0, making it easier to identify the center and axes of the ellipse.
3. What does the standard form tell us about the shape of an ellipse?
The standard form of an ellipse tells us that it is a symmetrical, closed curve with a center point and two axes, representing the widest and narrowest points of the ellipse. The ratio of a to b also indicates the elongation or flattening of the ellipse.
4. How do we graph an ellipse using the standard form?
To graph an ellipse using the standard form, we first plot the center point (h,k). Then, we move a units horizontally and b units vertically from the center to plot the endpoints of the semi-major and semi-minor axes. We can then use these points to sketch the ellipse.
5. Can we convert an ellipse from standard form to general form?
Yes, we can convert an ellipse from standard form to general form by expanding the squared terms and simplifying the equation. This can be useful when solving systems of equations or finding the equation of a tangent line to the ellipse.
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