Mathematics
Grade 11
15 min
Find the foci of an ellipse (Complete lesson above)
Find the foci of an ellipse (Complete lesson above)
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Introduction & Learning Objectives
Learning Objectives
Identify the major axis and orientation (horizontal or vertical) of an ellipse from its standard equation.
Determine the values of 'a', 'b', and the center (h, k) from the equation of an ellipse.
Calculate the focal distance 'c' using the relationship c^2 = a^2 - b^2.
Find the coordinates of the foci for an ellipse centered at the origin.
Find the coordinates of the foci for an ellipse centered at any point (h, k).
Distinguish between the formulas for finding foci on a horizontal versus a vertical ellipse.
Ever wondered how a 'whispering gallery' works, where a whisper on one side of a room can be heard clearly on the other? 🤫 The secret lies in the special points of an ellipse called the foci!
This lesson will gu...
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Key Concepts & Vocabulary
TermDefinitionExample
EllipseA flattened circle defined as the set of all points in a plane where the sum of the distances from two fixed points (the foci) is constant.The equation (x^2 / 16) + (y^2 / 9) = 1 represents an ellipse.
Foci (singular: Focus)The two fixed points inside an ellipse that are used to define its shape. They always lie on the major axis.If you tie a string to two pins (the foci) and trace a curve with a pencil held taut against the string, you will draw an ellipse.
Major AxisThe longest diameter of an ellipse, passing through its center and both foci. Its length is 2a.In the ellipse (x^2 / 25) + (y^2 / 16) = 1, the major axis is horizontal and has a length of 2 * sqrt(25) = 10.
Minor AxisThe shortest diameter of an ellipse, passing through the center and perpendicula...
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Core Formulas
Standard Equations of an Ellipse
Horizontal Major Axis: \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1 \\ Vertical Major Axis: \frac{(x-h)^2}{b^2} + \frac{(y-k)^2}{a^2} = 1
These are the standard forms for the equation of an ellipse centered at (h, k). The key is that a^2 is always the larger denominator, and it determines the orientation. If a^2 is under the x-term, the major axis is horizontal. If it's under the y-term, it's vertical.
Focal Distance Formula
c^2 = a^2 - b^2
This fundamental formula relates the semi-major axis (a), the semi-minor axis (b), and the focal distance (c). Use this to find 'c' after identifying a^2 and b^2 from the ellipse's equation. Remember, c = \sqrt{a^2 - b^2}.
Foci Coordinate Formulas
Horizontal Major Axis: Foci...
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Challenging
An ellipse has foci at (2, 5) and (10, 5). The length of its major axis is 10. What is the equation of the ellipse?
A.\frac{(x-6)^2}{25} + \frac{(y-5)^2}{9} = 1
B.\frac{(x-6)^2}{100} + \frac{(y-5)^2}{84} = 1
C.\frac{(x-5)^2}{25} + \frac{(y-6)^2}{9} = 1
D.\frac{(x-6)^2}{25} + \frac{(y-5)^2}{16} = 1
Challenging
An ellipse has foci at (-1, 8) and (-1, -2). The length of its minor axis is 8. Find the equation of the ellipse.
A.\frac{(x+1)^2}{16} + \frac{(y-3)^2}{25} = 1
B.\frac{(x+1)^2}{41} + \frac{(y-3)^2}{16} = 1
C.\frac{(x+1)^2}{16} + \frac{(y-3)^2}{41} = 1
D.\frac{(x-1)^2}{16} + \frac{(y+3)^2}{41} = 1
Challenging
Consider the ellipse \frac{(x-h)^2}{a^2} + \frac{(y-k)^2}{b^2} = 1. If the value of 'b' increases, while 'a' and the center (h, k) remain constant, what happens to the foci?
A.They move farther away from the center.
B.They move closer to the center.
C.They do not move.
D.They move to the minor axis.
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