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Questions and Answers
What is the definition of a focus in an ellipse?
What is the definition of a focus in an ellipse?
- The midpoint inside the ellipse curve.
- The points located outside the ellipse.
- Fixed points located between the center and the vertices. (correct)
- The endpoints of the major axis.
How is the length of the minor axis of an ellipse expressed?
How is the length of the minor axis of an ellipse expressed?
- 2b (correct)
- 2c
- 2a
- 2E
Which statement accurately describes the major axis of an ellipse?
Which statement accurately describes the major axis of an ellipse?
- It has endpoints that are the vertices of the ellipse. (correct)
- It is always shorter than the minor axis.
- It is the distance from a focus to the center.
- It divides the ellipse into two equal parts.
What is the eccentricity of an ellipse used to measure?
What is the eccentricity of an ellipse used to measure?
What is the significance of the directrix lines in relation to an ellipse?
What is the significance of the directrix lines in relation to an ellipse?
What is the value of $c$ in the equation derived from $c = a^2 - b^2$ if $a = 6$ and $b = 3$?
What is the value of $c$ in the equation derived from $c = a^2 - b^2$ if $a = 6$ and $b = 3$?
What are the coordinates of the vertices for the equation $\frac{x^2}{9} + \frac{y^2}{36} = 1$?
What are the coordinates of the vertices for the equation $\frac{x^2}{9} + \frac{y^2}{36} = 1$?
Given the formula for the foci as $F = (h, k \pm c)$, what is the value of $c$ calculated from the previously determined values $a$ and $b$?
Given the formula for the foci as $F = (h, k \pm c)$, what is the value of $c$ calculated from the previously determined values $a$ and $b$?
What are the coordinates of the center for the ellipse given by the equation $\frac{x^2}{9} + \frac{y^2}{36} = 1$?
What are the coordinates of the center for the ellipse given by the equation $\frac{x^2}{9} + \frac{y^2}{36} = 1$?
Which equation represents the alternate representation of the Left & Right endpoints of the L.R.?
Which equation represents the alternate representation of the Left & Right endpoints of the L.R.?
What does the variable $a$ equal in the earlier example if the graph of the ellipse has $a^2 = 36$?
What does the variable $a$ equal in the earlier example if the graph of the ellipse has $a^2 = 36$?
How are the Left and Right vertices connected to the center in an ellipse?
How are the Left and Right vertices connected to the center in an ellipse?
Determining $V_1$ gives what coordinate if calculated as $V_1 = rac{0}{6}$?
Determining $V_1$ gives what coordinate if calculated as $V_1 = rac{0}{6}$?
What are the coordinates of the center of the hyperbola represented by the equation?
What are the coordinates of the center of the hyperbola represented by the equation?
What is the value of 'a' in the given hyperbola equation?
What is the value of 'a' in the given hyperbola equation?
Which of the following correctly calculates the value of 'c' for the hyperbola?
Which of the following correctly calculates the value of 'c' for the hyperbola?
Which endpoints correspond to the vertices of the hyperbola?
Which endpoints correspond to the vertices of the hyperbola?
What is the horizontal principal axis of the hyperbola characterized by?
What is the horizontal principal axis of the hyperbola characterized by?
How is the value of the focal points (F1 and F2) calculated for this hyperbola?
How is the value of the focal points (F1 and F2) calculated for this hyperbola?
Which formula is used to find the length of the real axis (L.R.) for the hyperbola?
Which formula is used to find the length of the real axis (L.R.) for the hyperbola?
What does the term 'b' represent in the context of a hyperbola?
What does the term 'b' represent in the context of a hyperbola?
What is the standard equation of an ellipse with a horizontal major axis centered at the origin?
What is the standard equation of an ellipse with a horizontal major axis centered at the origin?
In the ellipse given by $\frac{x^2}{25} + \frac{y^2}{9} = 1$, what is the value of 'a'?
In the ellipse given by $\frac{x^2}{25} + \frac{y^2}{9} = 1$, what is the value of 'a'?
What are the coordinates of the foci for the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$?
What are the coordinates of the foci for the ellipse $\frac{x^2}{25} + \frac{y^2}{9} = 1$?
If an ellipse has its center at (h, k) and a horizontal major axis, how are the vertices calculated?
If an ellipse has its center at (h, k) and a horizontal major axis, how are the vertices calculated?
For the ellipse represented by $\frac{y^2}{16} + \frac{x^2}{9} = 1$, what is the value of 'b'?
For the ellipse represented by $\frac{y^2}{16} + \frac{x^2}{9} = 1$, what is the value of 'b'?
When calculating the endpoints of the latus rectum for an ellipse, which variables are used?
When calculating the endpoints of the latus rectum for an ellipse, which variables are used?
What is the relationship between the values of 'a', 'b', and 'c' for an ellipse?
What is the relationship between the values of 'a', 'b', and 'c' for an ellipse?
Which of the following describes the length of the latus rectum for an ellipse with a horizontal major axis?
Which of the following describes the length of the latus rectum for an ellipse with a horizontal major axis?
What are the coordinates of the center of the hyperbola described in the equation $\frac{(x + 3)^2}{64} - \frac{(y - 1)^2}{36} = 1$?
What are the coordinates of the center of the hyperbola described in the equation $\frac{(x + 3)^2}{64} - \frac{(y - 1)^2}{36} = 1$?
Which formula correctly represents the distance to the foci from the center of a hyperbola?
Which formula correctly represents the distance to the foci from the center of a hyperbola?
In the expression $\frac{(x + 1)^2}{25} + \frac{(y - 1)^2}{4} = 1$, what is the value of $a$?
In the expression $\frac{(x + 1)^2}{25} + \frac{(y - 1)^2}{4} = 1$, what is the value of $a$?
What is the general form of the hyperbola whose standard form is given by $\frac{(x + 3)^2}{64} - \frac{(y - 1)^2}{36} = 1$?
What is the general form of the hyperbola whose standard form is given by $\frac{(x + 3)^2}{64} - \frac{(y - 1)^2}{36} = 1$?
If the vertices of a hyperbola are given as $(-7.5, 5.29)$ and $(-7.5, -5.29)$, what is the length of the transverse axis?
If the vertices of a hyperbola are given as $(-7.5, 5.29)$ and $(-7.5, -5.29)$, what is the length of the transverse axis?
How would you determine the length of the conjugate axis for the hyperbola defined by the equation $\frac{(x + 3)^2}{64} - \frac{(y - 1)^2}{36} = 1$?
How would you determine the length of the conjugate axis for the hyperbola defined by the equation $\frac{(x + 3)^2}{64} - \frac{(y - 1)^2}{36} = 1$?
What are the coordinates of the foci for the hyperbola with center at $(-3, 0)$ and values $a = 8$, $b = 6$?
What are the coordinates of the foci for the hyperbola with center at $(-3, 0)$ and values $a = 8$, $b = 6$?
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Study Notes
Ellipse and Its Properties
- Definition: An ellipse is a closed curve formed by all points where the sum of the distances from the point to two fixed points (foci) is constant.
- Center: The midpoint of the ellipse, denoted by 'C'.
- Vertices: The endpoints of the major axis, denoted by 'V'.
- Foci: The two fixed points inside the ellipse that define its shape, denoted by 'F1' and 'F2'.
- Focal Distance: Distance between the center and one of the foci, denoted by 'c'.
- Directrix Lines: Lines outside of the ellipse, denoted by 'D.L.'
- Minor Axis: The shorter axis through the center, dividing the ellipse into symmetrical halves. Length = 2b.
- Major Axis (Principal Axis): The longer axis through the center, passing through the vertices. Length = 2a.
- Latus Rectum: A line segment passing through a focus, perpendicular to the major axis and extending to the ellipse. Length = 2(b^2/a).
- Eccentricity (E): The ratio of the focal distance to the semi-major axis (a). Eccentricity determines the shape of the ellipse (0 ≤ E ≤ 1).
- Standard Equations of the Ellipse:
- Center at Origin:
- Horizontal Major Axis: x^2/a^2 + y^2/b^2 = 1
- Vertical Major Axis: x^2/b^2 + y^2/a^2 = 1
- Center at (h, k):
- Horizontal Major Axis: (x - h)^2/a^2 + (y - k)^2/b^2 = 1
- Vertical Major Axis: (x - h)^2/b^2 + (y - k)^2/a^2 = 1
- Center at Origin:
Key Formulas
- Focal Distance: c = √(a^2 - b^2)
- Latus Rectum: L = 2(b^2/a)
- Eccentricity: E = c/a
Determining Parts of the Ellipse from Equation
- Center: The coordinates (h, k) from the standard form of the equation.
- Vertices: If major axis is horizontal: (h ± a, k); if vertical: (h, k ± a).
- Co-vertices: If major axis is horizontal: (h, k ± b); if vertical: (h ± b, k).
- Foci: If major axis is horizontal: (h ± c, k); if vertical: (h, k ± c).
- Latus Rectum: If major axis is horizontal: (h ± c, k ± b^2/a); if vertical: (h ± b^2/a, k ± c).
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