Mathematics
Grade 11
15 min
Find the foci of a hyperbola
Find the foci of a hyperbola
What you'll learn
- Solve subtraction problems up to three digits, and fill in the missing number to make both sides of the equation equal with 80% accuracy.
- Explain, using pictures or numbers, how to keep a subtraction equation balanced when changing a number on one side.
- Identify if a subtraction equation is balanced or not balanced after a change is made with 100% accuracy.
- Apply the concept of inverse operations (addition) to check if a subtraction equation is balanced with at least 75% accuracy.
Tutorial Preview
1
Introduction & Learning Objectives
Learning Objectives
Identify the orientation (horizontal or vertical) of a hyperbola from its standard equation.
Determine the values of 'a', 'b', and 'c' for a given hyperbola.
Apply the formula c^2 = a^2 + b^2 to find the focal distance.
Calculate the coordinates of the foci for a hyperbola centered at the origin (0,0).
Calculate the coordinates of the foci for a hyperbola with a shifted center (h, k).
Distinguish between the formulas for the foci of horizontal and vertical hyperbolas.
Ever wondered how GPS satellites pinpoint your location with such accuracy? 🛰️ The math behind it involves hyperbolas and their special 'focus' points!
This tutorial will guide you through the essential skill of finding the foci of a hyperbola. Unders...
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Key Concepts & Vocabulary
TermDefinitionExample
HyperbolaA type of conic section formed by the intersection of a double cone with a plane. It is the set of all points in a plane where the difference of the distances from two fixed points (the foci) is constant.The equation (x^2 / 9) - (y^2 / 16) = 1 represents a hyperbola.
Foci (plural of Focus)The two fixed points inside each curve of a hyperbola that are used to define it. The distance from any point on the hyperbola to the two foci has a constant difference.For the hyperbola (x^2 / 16) - (y^2 / 9) = 1, the foci are located at (5, 0) and (-5, 0).
Transverse AxisThe line segment that passes through the center and the two foci of the hyperbola. Its endpoints are the vertices of the hyperbola.For a horizontal hyperbola like (x^2 / a^2) - (y^2 / b^2) = 1, the transv...
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Core Formulas
Focal Distance Formula
c^2 = a^2 + b^2
This is the fundamental relationship between the distance to the vertex (a), the distance related to the conjugate axis (b), and the distance to the focus (c). Use this formula to find 'c' after identifying a^2 and b^2 from the hyperbola's standard equation.
Foci of a Horizontal Hyperbola
Foci are at (h ± c, k)
Use this for hyperbolas of the form ((x-h)^2 / a^2) - ((y-k)^2 / b^2) = 1. The foci lie on the horizontal transverse axis, so 'c' is added to and subtracted from the x-coordinate of the center (h).
Foci of a Vertical Hyperbola
Foci are at (h, k ± c)
Use this for hyperbolas of the form ((y-k)^2 / a^2) - ((x-h)^2 / b^2) = 1. The foci lie on the vertical transverse axis, so 'c' is added to...
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Challenging
The foci of a hyperbola are at (±10, 0) and its transverse axis has a length of 16. What is the equation of the hyperbola?
A.(x^2 / 64) - (y^2 / 36) = 1
B.(x^2 / 36) - (y^2 / 64) = 1
C.(x^2 / 256) - (y^2 / 100) = 1
D.(x^2 / 100) - (y^2 / 156) = 1
Challenging
A student analyzes ((y-1)^2 / 9) - ((x+4)^2 / 16) = 1. Their steps are: 1. Center is (-4, 1). 2. It's vertical. 3. a^2=9, b^2=16. 4. c^2 = 16 - 9 = 7. 5. Foci are at (-4, 1 ± √7). Which step contains the first conceptual error?
A.Step 1
B.Step 3
C.Step 4
D.Step 5
Easy
Which formula is used to find the focal distance, 'c', for any hyperbola with parameters 'a' and 'b'?
A.c^2 = a^2 - b^2
B.c^2 = a^2 + b^2
C.c = a + b
D.c^2 = b^2 - a^2
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What grade level is "Find the foci of a hyperbola"?
Find the foci of a hyperbola is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Find the foci of a hyperbola?
You'll be able to: Solve subtraction problems up to three digits, and fill in the missing number to make both sides of the equation equal with 80% accuracy; Explain, using pictures or numbers, how to keep a subtraction equation balanced when….
Is "Find the foci of a hyperbola" free to practice?
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How many practice questions are included with Find the foci of a hyperbola?
This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.