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Questions and Answers
What is the definition of a hyperbola?
What is the definition of a hyperbola?
The difference of the distances between the foci and a point on the hyperbola is fixed.
In hyperbolas, the sum of the distances between the foci and a point on the hyperbola is fixed.
In hyperbolas, the sum of the distances between the foci and a point on the hyperbola is fixed.
False (B)
What can the center of a hyperbola be?
What can the center of a hyperbola be?
- (h,k)
- Origin
- Both A and B (correct)
- None of the above
The standard equation of a hyperbola has two forms when the center is at the _____ .
The standard equation of a hyperbola has two forms when the center is at the _____ .
What are the transverse axis options for hyperbolas?
What are the transverse axis options for hyperbolas?
What is the formula for the coordinates of the vertices of a hyperbola centered at (h, k)?
What is the formula for the coordinates of the vertices of a hyperbola centered at (h, k)?
What is the general form of the standard equation of a hyperbola?
What is the general form of the standard equation of a hyperbola?
To convert general form to standard, you need to rearrange the equation: 4x^2 - 25y^2 - 24x - 64 = ____ .
To convert general form to standard, you need to rearrange the equation: 4x^2 - 25y^2 - 24x - 64 = ____ .
What is the result of converting the standard equation $(x + 2)^2/(4) - (y + 1)^2/(9) = 1$ to general form?
What is the result of converting the standard equation $(x + 2)^2/(4) - (y + 1)^2/(9) = 1$ to general form?
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Study Notes
Hyperbola Definition
- A hyperbola is a geometric shape where the difference of the distances between two fixed points (foci) and any point on the curve is constant.
Parts of a Hyperbola
- Center: Point where the axes of symmetry intersect.
- Transverse Axis: Line segment connecting the two vertices, it defines the opening direction of the hyperbola.
- Vertices: Points where the hyperbola intersects its transverse axis.
- Conjugate Axis: Line segment perpendicular to the transverse axis, its length is related to the distance between the foci.
- Foci: Two fixed points that define the hyperbola.
- Asymptotes: Lines that the hyperbola approaches as the distance from the center increases.
Equations of Hyperbola
-
Standard Form: The standard form of the equation simplifies the process of finding key features and graphing the hyperbola. There are two standard forms depending on the orientation of the transverse axis:
- Horizontal Transverse Axis: (x-h)^2/a^2 - (y-k)^2/b^2 = 1
- Vertical Transverse Axis: (y-k)^2/a^2 - (x-h)^2/b^2 = 1
-
General Form: The general form of the equation is a quadratic equation with both x and y terms. It needs to be rewritten in standard form for easier analysis.
Key Relationships
- Relationship between a, b, and c: c^2 = a^2 + b^2, where c is the distance from the center of the hyperbola to each focus.
- Vertices: (h±a, k) (horizontal axis), (h, k±a) (vertical axis)
- Co-vertices: (h±b, k) (horizontal axis), (h, k±b) (vertical axis)
- Foci: (h±c, k) (horizontal axis), (h, k±c) (vertical axis)
- Asymptotes: For hyperbolas centered at (h,k) and horizontal/vertical axes, the equations for the asymptotes are:
- Horizontal Axis: (y-k) = ±(b/a)(x-h)
- Vertical Axis: (x-h) = ±(b/a)(y-k)
Transformations and Graphing
- Shifting: The constants h and k represent horizontal and vertical shifts of the hyperbola's center from the origin.
- Stretching/Shrinking: The values of 'a' and 'b' determine the shape and orientation of the hyperbola.
Examples:
-
General to Standard:
- 4x^2 - 25y^2 - 24x - 64 = 0
- 3x^2 - y^2 + 18x + 4y + 35 = 0
-
Standard to General:
- x^2/25 - y^2/11 = 1
- (x+2)^2/4 - (y+1)^2/9 = 1
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