precalculusssssssssssssss (circles).pptx

INSTRUCTION: Read the questions
carefully and choose the letter with the
best answer. Write your answers in a
¼ clean sheet of paper.

1. Which of the following best describe the definition
of a circle?
a. It is a type of rectangle with four right angles and two pairs of
sides. One pair of parallel sides is longer than the other pair of
parallel sides.
b. It is collection of all the points in a plane, which are at a fixed
distance from a fixed point in the plane.
c. It is a one-dimensional locus of points that forms half of a 3600
arc.
d. It is a circular arc that measures 180° (equivalently, π radians, or
a half-turn).

2. Which among the following gives a
description of the measure of a
radius?
a. The radius is twice the measure of a
diameter.
b.The diameter is half the measure of a circle.
c. The diameter is half the measure of a radius.
d.The diameter is twice the measure of a

3. Which among the following gives a
measure of a diameter?
a. The radius is twice the measure of a
diameter.
b.The diameter is half the measure of a circle.
c. The diameter is half the measure of a radius.
d.The diameter is twice the measure of a
radius.

4. The following are the properties of a
circle EXCEPT one.
a. The distance from the centre of the circle to the
longest chord (diameter) is zero.
b. Circles having different radius are similar.
c. The radius of a circle is the longest chord of a
circle.
d. The radius drawn perpendicular to the chord
bisects the chord.

5. Which of the following state the
quadrant or axis that M (3,-2) lies in?
a. Quadrant I
b.Quadrant II
c. Quadrant III
d.Quadrant IV

quadrant or axis that C (1,0) lies in?
a. Quadrant I
b.Quadrant IV
c. y-axis
d.x-axis

quadrant or axis that S (7,2) lies in?
a. Quadrant I
b. Quadrant II
c. Quadrant III
d. Quadrant IV

8. Which of the following has the
correct formula in getting the
distance between the points?
a. d =
b. d=
c. d =
d. d =

9. What is the distance between two
points A and B whose coordinates are
(3, 2) and (9, 7), respectively?
a. unit
b. 15 units
c. 13 units
d. unit

10. Find the distance of a point P(4, 3)
from the origin.
a. 3 units
b.10 units
c. 7 units
d.5 units

Insufficient
Level
Fairly
Sufficient Level
Sufficient Level
0% - 40% 41% - 75% 76% - 100%

Copy
Cut!
Introduction to Conic Sections

INSTRUCTION: This activity is
allocated by pair. After performing
some instructions coming from the
teacher the learners will share their
observation/s on the activity to the
class.

a. Cut the
edges of printed
material B and
paste it into an
Oslo paper.

b. In printed
material B, cut
out the two holes
and the edges
using cutter or
scissors.

c. Cut out the
edges printed
material A and
glue the
semicircular
template into a
cone.

d. Place the plane
straight across the cone.

e. Place the plane at a
slant.

f. Place the plane parallel
to a side of a cone.

g. Place the in any
steeper of a cone.

Conic Sections:
Circles
Prepared by: Sir

Learning Goals
determine the
standard form of
equation of a
circle; and
Illustrate the different
types of conic sections:
parabola, ellipse, circle,
hyperbola, and
degenerate cases;
define a circle;
graph a circle in a
rectangular
coordinate
system.
01
03
02
04

Introduction to
Conic Sections
01

CONIC
SECTION
It is a curve formed
by the intersection of
a plane and a double
right circular cone.

A right circular cone
Generator
 It is a line lying entirely on
the cone.
Vertex
 It is an intersection where
all generators of a cone pass
through.
Nappe
 It is a lateral surface of a cone.

THREE TYPES OF CONIC SECTIONS
Ellipse
 If the cutting plane is not parallel to
any generator and intersects only one
cone to form a bounded curve.
5Circle
 If the cutting plane is not parallel to
any generator but is perpendicular to
the axis or horizontal.

Parabola
If the cutting plane is
parallel to one and only one
generator and intersects
only one cone to form an
unbounded curve.

Hyperbola
If the cutting plane is
parallel to two generators or
intersects both cones to
form two unbounded curves.

3Degenerate Conic Sections
(When the plane passes through the vertex)
Point Line
Two
Intersecting
Lines

1Conic Section
It is a set of points whose distances
from a fixed point are in constant
ratio to their distances from a fixed
line that is not passing through the
fixed point.

 In dealing with conic section, it is important to take note of the following
elements:
• focus (F) – the fixed point of the conic.
• directrix (d) - the fixed line d corresponding to the
focus.
• 2principal axis (a) – the line that passes through the
focus and perpendicular to the directrix. Every conic
is symmetric with respect to its principal axis.
• vertex (A) – the point of intersection of the conic
and its principal axis.
• eccentricity (e) – the constant ratio. If point P is one of
the points of the conic with point Q as its projection on
d, then the eccentricity is the ratio of the distance to the
distance , which is a constant. In symbols, e = .

The three types of conics are
distinguished by the value of its
eccentricity:
i. The conic is a parabola if the eccentricity e = 1.
ii.The conic is an ellipse if the eccentricity e < 1.
iii.The conic is a 4hyperbola if the eccentricity e >
1.

Definition of a
Circle and
Derivation of a
standard equation
of a Circle
02

Circle
It is a set of all coplanar points
such that the distance from a
fixed point is constant. The fixed
point is called the center of the
circle and the constant distance
from the center is called the
radius of the circle.

Circle
 Derive the equation of a circle
whose center C is at the point (0,0)
and with radius r, let P (x,y) be one
of the points of the circle.
• Derive the equation of a circle
whose center is at the point (h, k)
and with radius r.

Circle
Standard Form
(x – h)2
+ (y – k)2
= r2
x2
+ y2
= r2

Example
Determine the standard form of equation of the circle given its center and radius.
Draw its graph.
a. center C (0, 0), radius: 5 x2
+ y2
= r2
x2
+ y2
= 52
Since the center C (0, 0), then we
substitute the value of r in the
equation.
x2
+ y2
= 52 Square the value of r
x2
+ y2
= 25

Example
Draw its graph.
a. center C (-2, 7), radius: 4
(x + 2)2
+ (y – 7)2
= 16

Example
Draw its graph.
a. center C (-2, 7), radius: 4
Since the center C (h, k), then we
substitute all the given value to the
equation.
Square the value of r
(x + 2)2
+ (y – 7)2
= 16
(x – h)2
+ (y – k)2
= r2
(x + 2)2
+ (y – 7)2
= 42
(x + 2)2
+ (y – 7)2
= 16

Example
Draw its graph.
a. center C (0, 0), radius: 5
x2
+ y2
= 25

Examples
Draw its graph
a.center C (-8, -5), radius: 3
b.center C (, 2), radius:

Example
Find the center and radius of the circle given its standard form. Draw its
graph.
a. x2
+ y2
= 64
b. x2
+ y2
= 19
c. x2
+ (y – 2)2
= 49
d. (x + 7)2
+ (y – 5)2
= 56

Try me!
A. Determine the standard form of equation of the circle given its center
and radius. Draw its graph
a. center C (0, 0), radius: 2
b. center C (-12, -7), radius: 11
c. center C (, ), radius:

Try me!
B. Find the center and radius of the circle given its standard form. Draw
its graph.
a. x2
+ y2
= 9
b. (x + 0.8)2
+ (y – 0.6)2
= 1
c. (x - )2
+ (y – )2
=

DERIVE STANDARD FORM OF EQUATION
TO GENERAL FORM
(x – h)2
+ (y – k)2
= r2
x 2
– 2hx + h2
+ y2
– 2ky + k2
= r2
x 2
+ y2
– 2hx – 2ky + h2
+ k2
= r2
x 2
+ y2
– 2hx – 2ky + h2
+ k2
- r2
= 0
x 2
+ y2
+ (–2h)x + (–2k)y + (h2
+ k2
– r2
)= 0

THINK-PAIR-SHARE
Determine the general form of equation of the circle given its center and radius.
Draw its graph. Afterwards, discuss your answer to your pair/seatmate.
a. center R (2, -7), radius: 14
b. Think of any value of center and radius,
try to solve it and discuss your solution to
your partner.

Solving
situational
problems
involving circles
04

Given the line l containing two distinct points
P1 (x1, y1) and P2 (x2, y2) where x1 x2, then the
slope m of line l is computed as
• Slope of the line:
𝒎𝒍 =
𝒚 2 − 𝒚 1
𝒙 2 − 𝒙 1

The distance between two points P1 (x1, y1)
and P2 (x2, y2), on any xy plane is given as
Distance Formula:
𝒅=√(𝒙𝟐 − 𝒙𝟏 )
𝟐
+( 𝒚𝟐 − 𝒚𝟏 )
𝟐

EXAMPLES
1. Find the standard equation of the circle whose
diameter has endpoints (-5, 3) and (7, 11).
2. Find the standard and general equation of the circle
whose diameter has endpoints (-6, 2) and (8, 10).

TRY ME!
1.Find the equation of the circle that passes through the
points (2, 3) , (6, 1) , and (4, -3).

TRY ME!
1. A seismological station is located at (0, -4), 4km away from a
straight shoreline where the x-axis runs through. The
epicenter of an earthquake was determined to be 6km away
from the station.
a. Find the equation of the curve that contains the possible
location of epicenter.
b. If furthermore, the epicenter was determined to be 1km away
from the shore, find its possible coordinates (round off to two
decimal places)

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