**Conic Sections equations :**

**Conic Sections equations :**

__circle__A circle is the set of all points a given distance (the radius,

*r*) from a given point (the center). To get a circle from the right cones, the plane slice occurs parallel to the base of either cone, but does not slice through the element of the cones.

(x-a)^2 + (y-b)^2 = r^2

You can identify the equation for a circle when x and y are both squared, and the coefficients on them are the same — including the sign. the

*x*-squared and

*y*-squared have the same coefficient.

*An ellipse is the set of all points where the sum of the distances from two points (the foci) is constant, and you may be more familiar with the term oval. In order to get an ellipse from the two right cones, the plane must cut through one cone, not parallel to the base, and not through the element.*

__Ellipse__x^2/a^2+y^2/b^2=1

You can identify the equation for an ellipse when

*x*and

*y*are both squared and the coefficients are positive but different. The coefficients on

*x*-squared and

*y*-squared are different, but both are positive.

The distance from the center

*C*to either focus is

*f*=

*ae*, which can be expressed in terms of the major and minor radii:

f^2=a^-b^2

__Parabola__A

**parabola**is the set of points in a plane that are the same distance from a given point and a given line in that plane. The given point is called the

**focus,**and the line is called the

**directrix.**The midpoint of the perpendicular segment from the focus to the directrix is called the

**vertex**of the parabola. The line that passes through the vertex and focus is called the

**axis of symmetry**

The equation of a parabola can be written in two basic forms:

**Form 1:***y*=*a*(*x*–*h*)2 +*k***Form 2:***x*=*a*(*y*–*k*)2 +*h*

- In Form 1, the parabola opens vertically. (It opens in the “
*y*” direction.) If*a*> 0, it opens upward. Refer to Figure 1(a). If*a*< 0, it opens downward. The distance from the vertex to the focus and from the vertex to the directrix line are the same. This distance is 1/4a - A parabola with its vertex at (
*h*,*k*), opening vertically, will have the following properties. - The focus will be at(h,k+1/4a)
- The directrix will have the equation y=k-1/4a
- The axis of symmetry will have the equation
*x*=*h*. - Its form will be
*y*=*a*(*x*–*h*)2 +*k*.

- In Form 2, the parabola opens horizontally. (It opens in the “
*x*” direction.) If*a*> 0, it opens to the right. Refer to Figure 1(b). If*a*< 0, it opens to the left. - A parabola with its vertex at (
*h*,*k*), opening horizontally, will have the following properties. - The focus will be at (h+1/4a,k)
- The directrix will have the equation y=k-1/4a
- The axis of symmetry will have the equation
*y*=*k*. - Its form will be
*x*=*a*(*y*–*k*)2 +*h*.

__Hyperbola__

- A
**hyperbola**is the set of all points in a plane such that the absolute value of the difference of the distances between two fixed points stays constant. The two given points are the**foci**of the hyperbola, and the midpoint of the segment joining the foci is the**center**of the hyperbola. - A hyperbola has two axes of symmetry. The axis along the direction the hyperbola opens is called the
**transverse axis.**The**conjugate axis**passes through the center of the hyperbola and is perpendicular to the transverse axis. The points of intersection of the hyperbola and the transverse axis are called the**vertices**(singular,**vertex**) of the hyperbola. - A hyperbola centered at (0, 0) whose transverse axis is along the
*x*-axis has the following equation as its standard form.

- where (
*a*, 0) and (–*a*, 0) are the vertices and (*c*, 0) and (–*c*, 0) are its foci.

- As points on a hyperbola get farther from its center, they get closer and closer to two lines called
**asymptote lines.**The asymptote lines are used as guidelines in sketching the graph of a hyperbola. To graph the asymptote lines, form a rectangle by using the points (–*a*,*b*), (–*a*, –*b*), (*a*,*b*), and (*a*, –*b*) and draw its diagonals as extended lines. - For the hyperbola centered at (0, 0) whose transverse axis is along the
*x*-axis, the equation of the asymptote lines becomes