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Questions and Answers
What is the eccentricity of a parabola?
What is the eccentricity of a parabola?
- 1 (correct)
- 2
- 0.5
- 0
Which equation represents a parabola with its principal axis parallel to the x-axis?
Which equation represents a parabola with its principal axis parallel to the x-axis?
- $y - k = -4a(x - h)$
- $y - k = 4a(x - h)$ (correct)
- $y - k^2 = 4a(x - h)$
- $x - h = 4a(y - k)$
What defines the focus of a parabola?
What defines the focus of a parabola?
- It is the average distance of all points from the directrix.
- It is located halfway between the directrix and the vertex.
- It is the point from which the parabola opens. (correct)
- It is always placed at the origin of the graph.
What is the standard equation of a parabola that opens upward?
What is the standard equation of a parabola that opens upward?
What is the length of the latus rectum of a parabola?
What is the length of the latus rectum of a parabola?
Which of the following accurately describes the vertex of a parabola?
Which of the following accurately describes the vertex of a parabola?
Which parabola opens to the left based on its standard equation?
Which parabola opens to the left based on its standard equation?
Which components are essential parts of a parabola?
Which components are essential parts of a parabola?
What is the principal axis of the parabola described by the equation $x^2 - 20y = 0$?
What is the principal axis of the parabola described by the equation $x^2 - 20y = 0$?
What is the vertex of the parabola defined by the equation $x^2 - 20y = 0$?
What is the vertex of the parabola defined by the equation $x^2 - 20y = 0$?
What is the length of the latus rectum for a parabola with the equation $x^2 - 20y = 0$?
What is the length of the latus rectum for a parabola with the equation $x^2 - 20y = 0$?
How far is the focus from the vertex of the parabola given by $x^2 - 20y = 0$?
How far is the focus from the vertex of the parabola given by $x^2 - 20y = 0$?
What is the equation of the parabola that opens downward with vertex at $(h, k)$?
What is the equation of the parabola that opens downward with vertex at $(h, k)$?
Which of the following points represents an endpoint of the latus rectum for the parabola $x^2 - 20y = 0$?
Which of the following points represents an endpoint of the latus rectum for the parabola $x^2 - 20y = 0$?
In which direction does the focus of the parabola $x^2 - 20y = 0$ lie in relation to the vertex?
In which direction does the focus of the parabola $x^2 - 20y = 0$ lie in relation to the vertex?
What is the equation representing a parabola that opens to the left with vertex at $(h, k)$?
What is the equation representing a parabola that opens to the left with vertex at $(h, k)$?
What is the opening direction of the parabola defined by the equation $y^2 - 8x - 4y + 20 = 0$?
What is the opening direction of the parabola defined by the equation $y^2 - 8x - 4y + 20 = 0$?
What is the directrix of the parabola opening to the right with vertex at $(2, 2)$?
What is the directrix of the parabola opening to the right with vertex at $(2, 2)$?
What are the coordinates of the focus for the parabola with vertex at $(2, 2)$?
What are the coordinates of the focus for the parabola with vertex at $(2, 2)$?
What is the principal axis of symmetry for the parabola defined by $y^2 = 4ax$?
What is the principal axis of symmetry for the parabola defined by $y^2 = 4ax$?
What is the length of the latus rectum for the parabola with focus at $(4, 2)$?
What is the length of the latus rectum for the parabola with focus at $(4, 2)$?
Which of the following points are the endpoints of the latus rectum for the parabola with vertex $(2, 2)$?
Which of the following points are the endpoints of the latus rectum for the parabola with vertex $(2, 2)$?
What is the standard equation of a parabola opening to the right with vertex at the origin (0, 0)?
What is the standard equation of a parabola opening to the right with vertex at the origin (0, 0)?
What value of $a$ is determined for the parabola that passes through the point (1, -2) with vertex at the origin and opening right?
What value of $a$ is determined for the parabola that passes through the point (1, -2) with vertex at the origin and opening right?
What form does the standard equation of a downward-opening parabola take?
What form does the standard equation of a downward-opening parabola take?
What is the distance 'a' for the parabola with vertex at (3, 1) and focus at (3, -5)?
What is the distance 'a' for the parabola with vertex at (3, 1) and focus at (3, -5)?
What will the equation of the parabola be if the vertex is located at (6, -3) and it opens upwards with a latus rectum length of 20?
What will the equation of the parabola be if the vertex is located at (6, -3) and it opens upwards with a latus rectum length of 20?
If the focus of a parabola that opens right is given as (-3, 4), what would be the equation of its principal axis?
If the focus of a parabola that opens right is given as (-3, 4), what would be the equation of its principal axis?
What is the correct general equation for a parabola with focus at (-2, -3) and directrix $x = 3$?
What is the correct general equation for a parabola with focus at (-2, -3) and directrix $x = 3$?
In the provided content, the vertex of the parabola with endpoints of latus rectum at (-4, 2) and (16, 2) is located at:
In the provided content, the vertex of the parabola with endpoints of latus rectum at (-4, 2) and (16, 2) is located at:
Which statement is true regarding the parabola with the focus located at (3, -5)?
Which statement is true regarding the parabola with the focus located at (3, -5)?
What is the form of the parabolic equation for a parabola that opens to the left?
What is the form of the parabolic equation for a parabola that opens to the left?
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Study Notes
Parabola Definition
- A parabola is a conic section whose eccentricity is 𝑒 = 1.
- It is defined as the locus of a point that moves in a plane so that its distance from a fixed point (Focus) is equal to its distance from a fixed line (Directrix).
Standard Form of the Equation
- Equation of Parabola with Principal Axis Parallel to x-axis:
- Vertex at 𝑉(ℎ, 𝑘) and opens to the right: 𝒚−𝒌𝟐 = 𝟒𝒂 𝒙−𝒉
- Vertex at 𝑉(ℎ, 𝑘) and opens to the left: 𝒚−𝒌𝟐 = −𝟒𝒂 𝒙−𝒉
- Equation of Parabola with Principal Axis Parallel to y-axis:
- Vertex at 𝑉(ℎ, 𝑘) and opens upward: 𝒙−𝒉𝟐 = 𝟒𝒂 𝒚−𝒌
- Vertex at 𝑉(ℎ, 𝑘) and opens downward: 𝒙−𝒉𝟐 = −𝟒𝒂 𝒚−𝒌
Key Components
- Principal Axis/Axis of Symmetry/Axis of Parabola: Perpendicular line to the directrix passing through the focus.
- Latus Rectum: The chord through the focus perpendicular to the principal axis.
- Vertex: Points that cut through the principal axis.
- Length of Latus Rectum: 4𝑎, where 𝑎 is the directed distance from the vertex to the focus.
Determining the Opening
- Opens to the Right: If 𝑎 > 0 and the principal axis is parallel to x-axis.
- Opens to the Left: If 𝑎 < 0 and the principal axis is parallel to x-axis.
- Opens Upward: If 𝑎 > 0 and the principal axis is parallel to y-axis.
- Opens Downward: If 𝑎 < 0 and the principal axis is parallel to y-axis.
General Equation of Parabola
- The general equation of a parabola with a principal axis parallel to the y-axis is 𝑨𝒙𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎.
- The general equation of a parabola with a principal axis parallel to the x-axis is 𝐁𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎.
Finding the Equation
- Example: Find the equation of the parabola with vertex at the origin, axis at the x-axis, and passing through (1,-2).
- Steps:
- Determine the opening of the parabola based on the given information.
- Use the standard equation corresponding to the opening.
- Substitute the vertex coordinates (ℎ, 𝑘) into the standard equation.
- Substitute the point (1, -2) into the equation and solve for 𝑎.
- Write the final equation with the value of 𝑎.
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