Mathematics
Grade 11
15 min
Write equations of hyperbolas in standard form using properties
Write equations of hyperbolas in standard form using properties
What you'll learn
- Solve subtraction problems with four-digit numbers, regrouping when needed, and get the correct answer at least 8 out of 10 times.
- Explain the steps you take to subtract two four-digit numbers, including when and why you need to regroup, in your own words.
- Identify when a subtraction problem requires regrouping by looking at the digits in each place value.
- Apply subtraction skills to solve word problems involving four-digit numbers, showing your work and getting the correct answer at least 3 out of 5 times.
Tutorial Preview
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Introduction & Learning Objectives
Learning Objectives
Identify the orientation (horizontal or vertical) of a hyperbola from its given properties.
Determine the center (h, k) of a hyperbola using the midpoint of its vertices or foci.
Calculate the values of 'a' and 'c' based on the coordinates of the center, vertices, and foci.
Use the relationship c² = a² + b² to find the value of b².
Write the standard form equation for a horizontal hyperbola given its properties.
Write the standard form equation for a vertical hyperbola given its properties.
Derive the complete equation of a hyperbola from a set of properties like vertices and foci.
Ever wonder how GPS pinpoints your location or how a cooling tower gets its unique curve? 🛰️ The secret lies in the powerful geometry of the hyperbola!
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Key Concepts & Vocabulary
TermDefinitionExample
Center (h, k)The midpoint of the line segment connecting the foci, and also the midpoint of the line segment connecting the vertices. It is the point of symmetry for the hyperbola.If the vertices are at (1, 5) and (9, 5), the center is at ((1+9)/2, (5+5)/2) = (5, 5).
VerticesThe two points where the hyperbola intersects its transverse axis. They are the turning points of the two branches.For the hyperbola (x²/9) - (y²/16) = 1, the vertices are at (-3, 0) and (3, 0).
FociTwo fixed points on the interior of each curve of the hyperbola that define its shape. The difference of the distances from any point on the hyperbola to the two foci is constant.For a hyperbola with vertices at (±4, 0) and b=3, the foci are at (±5, 0) because c² = 4² + 3² = 25, so c=5.
Transverse Axi...
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Core Formulas
Standard Form: Horizontal Transverse Axis
\frac{(x-h)^2}{a^2} - \frac{(y-k)^2}{b^2} = 1
Use this form when the vertices and foci lie on a horizontal line (i.e., they have the same y-coordinate). The x-term is the positive term.
Standard Form: Vertical Transverse Axis
\frac{(y-k)^2}{a^2} - \frac{(x-h)^2}{b^2} = 1
Use this form when the vertices and foci lie on a vertical line (i.e., they have the same x-coordinate). The y-term is the positive term.
Focus-Vertex Relationship
c^2 = a^2 + b^2
This crucial formula connects the distances 'a', 'b', and 'c'. Use it to find the value of b² when you know 'a' and 'c'. Note that for a hyperbola, c is always greater than a.
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Challenging
A hyperbola has vertices at (3, 3) and (9, 3). One of its asymptotes has the equation y - 3 = (2/3)(x - 6). What is the standard form equation of the hyperbola?
A.\frac{(x-6)^2}{9} - \frac{(y-3)^2}{36} = 1
B.\frac{(y-3)^2}{9} - \frac{(x-6)^2}{4} = 1
C.\frac{(x-6)^2}{9} - \frac{(y-3)^2}{4} = 1
D.\frac{(x-6)^2}{4} - \frac{(y-3)^2}{9} = 1
Challenging
The foci of a hyperbola are at (-2, 1) and (8, 1), and one of its asymptotes has a slope of 3/4. Find the equation of the hyperbola.
A.\frac{(x-3)^2}{9} - \frac{(y-1)^2}{16} = 1
B.\frac{(x-3)^2}{16} - \frac{(y-1)^2}{9} = 1
C.\frac{(y-1)^2}{16} - \frac{(x-3)^2}{9} = 1
D.\frac{(x-3)^2}{25} - \frac{(y-1)^2}{14.0625} = 1
Challenging
A hyperbola's center is the midpoint of the line segment from (-2, 8) to (4, 2). Its transverse axis is vertical and has a length of 10 units. The distance from the center to each focus is √34 units. What is the hyperbola's equation?
A.\frac{(y-5)^2}{100} - \frac{(x-1)^2}{9} = 1
B.\frac{(x-1)^2}{25} - \frac{(y-5)^2}{9} = 1
C.\frac{(y-5)^2}{25} - \frac{(x-1)^2}{34} = 1
D.\frac{(y-5)^2}{25} - \frac{(x-1)^2}{9} = 1
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Write equations of hyperbolas in standard form using properties is a Grade 11 Mathematics lesson on ExcelOS.
What will I learn in Write equations of hyperbolas in standard form using properties?
You'll be able to: Solve subtraction problems with four-digit numbers, regrouping when needed, and get the correct answer at least 8 out of 10 times; Explain the steps you take to subtract two four-digit numbers, including when and why you need to….
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This lesson includes 25 practice questions across multiple difficulty levels, each with instant feedback and explanations.