What is a conic that consists all points in the plane equidistant from a fixed point called focus F and a fixed line l called Directrix not containing F *?

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.

• The line over which a parabola is symmetric.

• The term for each of the two distinct sections of the graph of a hyperbola.

• 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.

• The set of all points equidistant from a given fixed point.

• The intersection of a plane and a right circular cone.

• The line segment related to a hyperbola of length 2b whose midpoint is the center.

• A conic which is not a parabola, ellipse, circle, or hyperbola. These include lines, intersecting lines, and points.

• A line segment that contains the center of a circle whose endpoints are both on the circle, or sometimes, the length of that segment.

• 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.

• 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.

• 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.

• 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.

• The line segment containing the foci of an ellipse whose endpoints are the vertices whose length is 2a.

• The line segment containing the center of an ellipse perpendicular to the major axis whose length is 2b.

• 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.

• A segment between the center of a circle and a point on the circle, or sometimes, the length of that segment.

• The line segment that contains the center and whose endpoints are the two vertices of a hyperbola.

• (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 = , orr = , where e is the eccentricity of the conic, the pole is the focus, and p is the distance between the focus and the directrix.

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: + = 1. The center is at (h, k). The length of the major axis is 2a, and the length of the minor axis is 2b. The distance between the center and either focus is c, where c2 = a2 - b2. a > b > 0. The standard equation of an ellipse with a vertical major axis is the following: + = 1. The center is at (h, k). The length of the major axis is 2a, and the length of the minor axis is 2b. The distance between the center and either focus is c, where c2 = a2 - b2. a > b > 0.

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.