A parabola is set of all points in a plane which are an equal distance away from a given point and given line. The point is called the focus of the parabola, and the line is called the directrix .
The directrix is perpendicular to the axis of symmetry of a parabola and does not touch the parabola. If the axis of symmetry of a parabola is vertical, the directrix is a horizontal line .
If we consider only parabolas that open upwards or downwards, then the directrix is a horizontal line of the form y = c .
Relation between focus, vertex and directrix:
The vertex of the parabola is at equal distance between focus and the directrix.
If F is the focus of the parabola, V is the vertex and D is the intersection point of the directrix and the axis of symmetry, then V is the midpoint of the line segment F D ¯ .
Example:
If a parabola has a vertical axis of symmetry with vertex at ( 1 , 4 ) and focus at ( 1 , 2 ) , find the equation of the directrix.
If F is the focus of the parabola, V is the vertex and D is the intersection point of the directrix and the axis of symmetry, then V is the midpoint of the line segment F D ¯ .
Equate the x -coordinates and solve for p .
1 = 1 + p 2 2 = 1 + p p = 1
Equate the y -coordinates and solve for q .
4 = 2 + q 2 8 = 2 + q q = 6
The equation of the directrix is of the form y = c and it passes through the point ( 1 , 6 ) . Here, c = 6 .
So, the equation of the directrix is y = 6 .
The graph is as shown.
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The line over which a parabola is symmetric.
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The term for each of the two distinct sections of the graph of a hyperbola.
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For an ellipse and hyperbola, the midpoint between the foci. For a circle, the fixed point from which all points on the circle are equidistant.
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The set of all points equidistant from a given fixed point.
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The intersection of a plane and a right circular cone.
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The line segment related to a hyperbola of length 2b whose midpoint is the center.
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A conic which is not a parabola, ellipse, circle, or hyperbola. These include lines, intersecting lines, and points.
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A line segment that contains the center of a circle whose endpoints are both on the circle, or sometimes, the length of that segment.
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For a parabola, it is the line whose distance from any point on the parabola is the same as the distance from that point to the focus. For a conic defined in polar terms, it is the line whose distance from any point on the conic makes a constant ratio with the distance between that point and the focus.
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The ratio
in an ellipse or hyperbola. Under the polar definition of conics, e is the constant ratio of the distance from a point to the focus and the distance from that point to the directrix. -
The set of all points such that the sum of the distances from the point to each of two fixed points is constant.
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For a parabola, the point whose distance from any point on the parabola is the same as the distance between that point and the directrix. For an ellipse, one of two points--the sum of whose distances to a point on the ellipse is constant. For a hyperbola, one of two points--the difference of whose distances to a point on the hyperbola is constant. Under the polar definition of a conic, it is the point whose distance from a point on the conic makes a constant ratio with the distance between that point and the directrix.
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The set of all points such that the difference of the distances between each of two fixed points and any point on the hyperbola is constant.
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The line segment containing the foci of an ellipse whose endpoints are the vertices whose length is 2a.
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The line segment containing the center of an ellipse perpendicular to the major axis whose length is 2b.
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The set of all points such that the distance between a point on the parabola and a fixed line is the same as the distance between a point on the parabola and a fixed point.
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A segment between the center of a circle and a point on the circle, or sometimes, the length of that segment.
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The line segment that contains the center and whose endpoints are the two vertices of a hyperbola.
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(Plural = "vertices") For a parabola, the point halfway between the focus and the directrix. For an ellipse, one of two points where the line that contains the foci intersects the ellipse. For a hyperbola, one of two points at which the line containing the foci intersects the hyperbola.
Polar Form of a Conic | r = |
Standard Form of a Circle | The standard equation for a circle is (x - h)2 + (y - k)2 = r2. The center is at (h, k). The radius is r. |
Standard Form of an Ellipse | The standard equation of an ellipse with a horizontal major axis is the following: |
Standard Form of a Hyperbola | The standard equation for a hyperbola with a horizontal transverse axis is - = 1. The center is at (h, k). The distance between the vertices is 2a. The distance between the foci is 2c. c2 = a2 + b2. The standard equation for a hyperbola with a vertical transverse axis is - = 1. The center is at (h, k). The distance between the vertices is 2a. The distance between the foci is 2c. c2 = a2 + b2. |
Standard Form of a Parabola | If a parabola has a vertical axis, the standard form of the equation of the parabola is this: (x - h)2 = 4p(y - k), where p≠ 0. The vertex of this parabola is at (h, k). The focus is at (h, k + p). The directrix is the line y = k - p. The axis is the line x = h. If a parabola has a horizontal axis, the standard form of the equation of the parabola is this: (y - k)2 = 4p(x - h), where p≠ 0. The vertex of this parabola is at (h, k). The focus is at (h + p, k). The directrix is the line x = h - p. The axis is the line y = k. |