The document explains conic sections, which are figures formed by slicing a double cone with a plane, including definitions and properties of circles, parabolas, ellipses, and hyperbolas. Each conic section has a specific standard equation and distinct characteristics, such as the constant distance in ellipses and hyperbolas. The document also provides equations for deriving these shapes based on given parameters.

1. CONIC SECTIONS
By: Jasmin Joyce M. Terrado
Candice P. Madrid

2. DEFINITIONS
Conic Section: Any figure that can be formed by slicing a double cone with a
plane
Parabola Circle Ellipse Hyperbola

3. EQUATION OF A CONIC SECTION
2 2
0
where A, B, and C are not all zero.
Ax Bxy Cy Dx Ey F     

4. DISTINCT PROPERTIES OF CONIC SECTIONS
 Parabola: A = 0 OR C = 0
 Circle: A = C
 Ellipse: , but both have the same sign
 Hyperbola: A and C have Different signs
A C

5. 1. CIRCLE
A circle is a simple closed shape. It is the set of all points in a plane that are at a given distance from a
given point, the centre; equivalently it is the curve traced out by a point that moves so that its distance from
a given point is constant.
Standard Form: x² + y² = r²
You can determine the equation for a circle by using
the distance formula then applying the standard form
equation.
Or you can use the standard form.
Most of the time we will assume the center is (0,0). If
it is otherwise, it will be stated.
It might look like: (x-h)² + (y – k)² = r²

6. II. PARABOLA
A parabola is a curve where any point is at an equal distance from: a fixed point (the focus ), and. a fixed
straight line (the directrix )
STANDARD EQUATION OF A PARABOLA:
Let the vertex be (h, k) and p be the distance between the
vertex and the focus and p ≠ 0.
(x−h)2=4p(y−k) (x−h)2=-4p(y−k)vertical
axis;
directrix is y = k - p
(y−k)2=4p(x−h) (y−k)2=-
4p(x−h) horizontal axis;
directrix is x = h - p

7. III. ELLIPSE
An ellipse is an important conic section and is formed by intersecting a cone with a plane that does not go
through the vertex of a cone. The ellipse is defined by two points, each called a focus. From any point on
the ellipse, the sum of the distances to the focus points is constant. The position of the foci determine the
shape of the ellipse.
STANDARD EQUATION OF AN ELLIPSE:

8. IV. 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.