This document provides information about circles and parabolas. It defines a circle as all points equidistant from a fixed center point and gives the standard form of a circle equation. Examples are provided of writing the equation of a circle given its center and radius. The document then defines a parabola as all points equidistant from a fixed focus point and directrix line and gives the basic forms of parabola equations. Examples are worked through of finding the focus, directrix, and sketching graphs of parabolas given their equations.

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2. 2nd 4th 5th
1st 3rd
Pre- Calculus
Week 1-2
Tchr. James Luck M. Valenzuela

3. Circle is one the conic sections
which were discovered during the
classical Greek Period, 600 to 300
BC. These are the intersections of a
plane and a double-napped cone
(you will learn more of these in Pre-
Calculus in the senior high school
level)

4. We should begin by defining conics in terms
of the intersections of planes and cone, as the
Greeks did, or we could define them algebraically in
terms of the general second-degree equation.
Ax2 + Bx + Cy2 + Dx + Ey + F = 0

5. However, we will study a third approach,
in which each of the conics is defined as a
locus (collection) of points satisfying a certain
geometric property. For example, the definition
of a circle as the collection of all points (x,y)
that are equidistant from a fixed point (h,k)
leads to the standard equation of the circle.
(x – h)2 + (y-k)2 = r2

6. CIRCLES
Is the set of all points (x,y) in
a plane that are equidistant to
a fixed point (h, k) which is
known as center of the circle.
The distance between the
center and any point is known
as radius of the circle.
The distance Formula can be
used to obtain an equation of
a circle whose center is at (h,
k) and whose radius is r.
Center
(h,k)
A point on
the circle
(x, y)

7. STANDARD FORM OF THE
EQUATION OF A CIRCLE
(x – h)2 + (y - k)2 = r2
The point (h, k) is the center of the
circle, and the positive number r is the
radius of the circle.
However, if the center is at the origin
(h, k) = (0, 0), the standard equation
would be
x2 + y2 = r2

8. FINDING THE STANDARD EQUATION
OF THE CIRCLE.
Example 1. Find the equation of the circle with center at the
origin with radius 3.

9. FINDING THE STANDARD EQUATION
OF THE CIRCLE.
Example 2. Find the equation of the circle with center at (2, 3)
with radius 5.

10. FINDING THE STANDARD EQUATION
OF THE CIRCLE.
Example 3. Find the standard equation of the circle with
endpoints of the diameter at (-6, -2) and (0, -2)

11. GRAPHING CIRCLES GIVEN
ITS EQUATION
Graphing circles is easy if you already
have the coordinates of the center and the
radius.

12. GRAPHING CIRCLES
Example 1. Graph the circle x2 + y2 = 9

13. GRAPHING CIRCLES
Example 2. Graph the circle with equation
x2 + y2 + 10x – 2y = 55.

14. Let’s PRACTICE!

15. A. Solve for the standard
equation of each circle
1. Center at origin, r = 8
2. Has a diameter with endpoints
(-1, 4) and (4,2)
B. Solve for the general
equation of each circle
1. Center (- 4, 3) r= 13
2. Center (5, - 6) tangent to the
y-axis
C. Graph each circle
1. x2 + y2 = 16
2. x2 + y2 +4x + 8y = - 4

16. THE PARABOLA

17. The three-point line represents the
parabola which is a set of all points
equidistant to a fixed point known as focus
which the middle of the free throw line
represent and to a fixed line known as
directrix which is represented by the half
court line although the directrix is closer to
the focus compared to the actual
basketball court.

18. In this chapter, we restrict our attention to
parabola that are situated with the vertex at
the origin and that have a vertical or
horizontal axis of symmetry. If the focus of
such parabola is the point F (o,p), then the
axis of symmetry must be vertical, and the
directrix has the equation y=-p. The figure
at the right illustrates the case p>0.
If p > 0, then the parabola opens upward;
but if p< 0, it opens downward. When the
value of x coordinate is changed to –x, the
equation remains unchanged, so the graph
is symmetric about the y-axis.

19. EQUATIONS AND GRAPHS OF
PARABOLAS

20. EXAMPLE 1. Find the equation for the
parabola with vertex V(0,0) and focus
F(0,2) and sketch its graph.
PARABOLA WITH VERTICAL AXIS
x2 = 4py ; vertex = V(0,0 Focus F(0,p)
Directrix (y=-p)
The parabola opens upward if p>0 and
downward if p<0.

21. EXAMPLE 2. Find the focus and directrix
of the parabola y = -x2 and sketch its graph.
PARABOLA WITH VERTICAL AXIS
x2 = 4py ; vertex = V(0,0 Focus F(0,p)
Directrix (y=-p)
The parabola opens upward if p>0 and
downward if p<0.

24. EXAMPLE 3. A parabola has the equation
6x + y2 =0. Find the focus and directix of
the parabola and sketch the graph.
PARABOLA WITH HORIZONTAL AXIS
y2 = 4px ; vertex = V(0,0) Focus
F(0,p) Directrix (x=-p)
The parabola opens to the right if p>0 and
to the left if p<0.