San Miguel Academy Inc.
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LEARNING MODULE
Pre-Calculus
Name: __________________________________
Grade 11 | Q1
Module 1
Year & Section: ______________________
Chapter 1: Conic Section
Lesson Objectives
At the end of this lesson, you are expected to, to illustrate the different types of conic
sections: parabola, ellipse, circle, hyperbola, and degenerate cases.
Subtasks:
Define the different types of conics: parabola, ellipse, circle, hyperbola, and degenerate
cases.
✔ Starting Point
What is conic? What are the different types of conics?
A Conic is the intersection of a plane and a right circular cone. The four basic types of conics are
parabolas, ellipses, circles, and hyperbolas.
How do you distinguish one from another?
 The determinant of the equation is 𝑏 2 − 4𝑎𝑐. Assuming a conic is not degenerate. The
following conditions hold true:
𝟐
If 𝒃 − 𝟒𝒂𝒄 > 𝟎, the conic is a hyperbola.
If 𝒃𝟐 − 𝟒𝒂𝒄 < 𝟎, the conic is a circle or an ellipse.
If 𝒃𝟐 − 𝟒𝒂𝒄 = 𝟎, the conic is a parabola
Another way to classify conics has to do with the product of A and C. Assuming a conic is not
degenerate, the following conditions hold true: if AC > 0, the conic is an ellipse or a circle. If AC
< 0, the conic is a hyperbola. If AC = 0, and A and C are not both zero, the conic is a parabola,
Finally , if A = C, the conic is a circle.
In the conics at the right side, the plane does not pass
through the vertex of the cone. When the plane does
intersect the vertex of the cone, the resulting conic is
called a degenerate Conic. Degenerate conics include
a point, a line, and two intersecting lines.
The equation of every conic can be written in the
following.
Form: 𝐴𝑥 2 + 𝐵𝑥𝑦 + 𝐶𝑦 2 + 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0. This is
the algebraic definition of a conic. Conics can be
classified according to the coefficient of this equation.
Circle is the set of all points P in a plane that are the same distance from a given point. The
given distance is the radius of the circle, and the given point is the center of the circle. Below is
the equation of the circle, sample equation and its graph.
Parabola is the set of all points P in a plane that are the same distance from a given line and a
fixed point not on the line. The fixed point is called the focus and the given line is called the
directrix of the parabola. The parabola may open upward, downward, to the right, or to the left.
Below is the equation of the parabola, sample equation and its graph
Ellipse is the set of all points P in a plane such that the sum of the distances from P to two fixed
points is a constant. Each of the fixed points is a focus. Below is the equation of the ellipse,
sample equation and its graph.
Hyperbola is the set of all points P in a plane such that the differences of the distances from P to
two fixed points are a constant. The two fixed points are called foci. Below is the equation of the
hyperbola, sample equation and its graph.
Lesson Objectives
At the end of this lesson, you are expected to, to define a circle, determine the standard
form of equation of a circle, and graph a circle in a rectangular coordinate system.
Subtasks:
Determine the center and radius of a Circle, Identify the radius from a given Diameter
using Midpoint Theorem, and Solve for the distance between two given points.
✔ Starting Point
The conic section formed when a cone is cut by a plane parallel to its base (with this plane not
passing through the cone’s vertex), is a circle. We can define a circle analytically:
A Circle is the set of all points having the same distance (the radius) from a given point
(the center)
Developing Standard Equation of a Circle
Example 1: Determine the equation of a circle with center C (3, 2) and radius r = 5.
Answer:
(x − h)2 + (y − k)2 = 𝑟 2
(x − 3)2 + (y − 2)2 = 52
(x − 3)2 + (y − 2)2 = 25
Substitute the values of h and k from C(3,2)
Simplify 52 = 25.
Example 2: Determine the equation of a circle with center C (-5, 2) and passing through the
point P(-8, 7).
Answer:
(x − h)2 + (y − k)2 = 𝑟 2
(x − (−5))2 + (y − 2)2 = 𝑟 2
(x + 5)2 + (y − 2)2 = 𝑟 2
Substitute the values of h and k from C(-5, 2)
Use the point P(-8, 7) to find the value or r 2
(x + 5)2 + (y − 2)2 = 𝑟 2
(−8 + 5)2 + (7 − 2)2 = 𝑟 2
(−3)2 + (5)2 = 𝑟 2
9 + 25 = 𝑟 2
34 = 𝑟 2
Therefore, the equation of the circle in standard form is
(𝐱 + 𝟓)𝟐 + (𝐲 − 𝟐)𝟐 = 𝟑𝟒
Example 3: By looking on the graph, express the equation of the circle in standard form.
Example 4: What is the standard form of the equation of a circle with endpoints of a diameter at
the points (7,8) and (-5,6)?
Midpoint Formula:
𝒙 𝟏 + 𝒙 𝟐 𝒚𝟏 + 𝒚𝟐
(
,
)
𝟐
𝟐
Answer:
Center of the circle is the midpoint of the diameter,
7−5 8+6
( 2 , 2 ) = (1, 7)
Again, the diameter is the distance between the points (7, 8)
and (-5, 6):
Distance Formula:
√(𝒙𝟐 − 𝒙𝟏 )𝟐 + (𝒚𝟐 − 𝒚𝟏 )𝟐
2
√(7 − (−5)) + (8 − 6)2
= √(12)2 + (2)2
= 2√37
Example 5. Find the center and radius of the circle having the equation:(x − 5)2 + (y − 8)2 =
81
Answer:
(x − h)2 + (y − k)2 = 𝑟 2
Using this form, center is the point (h, k) and radius is “r” h = 5, k = 8, r = 9
Therefore, the center is point (5, 8) and the radius is 9.
Example 6. Find the center and radius of the circle having the equation: 4𝑥 2 − 4𝑦 2 − 16𝑥 −
24𝑦 + 51 = 0
Answer:
Graphing a Circle in the Coordinate Plane
The equation of a circle appears as (x − h)2 + (y − k)2 = 𝑟 2. This is called the standard form
(or the center-radius form). The h and k represent the center of the circle at point (h, k), and r
names the radius. Specifically, h represents the horizontal displacement — how far to the left or
to the right the center of the circle falls from the y-axis. The variable k represents the vertical
displacement — how far above or below the center falls from the x-axis. We can count from the
center r units (the radius) horizontally in both directions and vertically in both directions. This
will give you four different points, all equidistant from the center. Connect these four points with
the best curve that you can sketch to get the graph of the circle.
Example A. Center at the Origin
Graph the circle: : 𝑥 2 + 4𝑦 2 = 16
Follow the steps:
1. Realize that the circle is centered at the origin (no h and k) and
place this point there.
2. Calculate the radius by solving for r. Set r-squared = 16. In this
case, you get r = 4.
3. Plot the radius points on the coordinate plane. You count out 4
in every direction from the center (0, 0): left, right, up, and down.
4. Connect the dots to graph the circle using a smooth, round
curve.
Example B. Center away from the Origin
Graph the circle: : (𝑥 − 3)2 + (𝑦 − 1)2 = 25
Follow the steps:
1. Locate the center of the circle from the equation (h, k) Place
the center of the circle at (3, -1) 2. Calculate the radius by solving
for r. Set r-squared = 25. In this case, you get r = 5.
3. Plot the radius points on the coordinate plane. Count 5 units
left, right, up, and down from the center at (3, -1). This means
that you should have points at (8, -1), (-2, -1), (3, -6), and (3, 4).
4. Connect the dots to graph the circle using a smooth, round
curve.
Lesson Objectives
At the end of this lesson, you are expected to, to determine the standard form of equation
of a parabola.
Subtasks:
Define a parabola, and graph a parabola in a rectangular coordinate system.
✔ Starting Point
The parabola is the curve formed from all the points (x, y) that are equidistant from the directrix
and the focus. The line perpendicular to the directrix and passing through the focus (that is, the line that
splits the parabola up the middle) is called the “axis of symmetry”. The point on this axis which is
midway between the focus and the directrix in the “vertex”, the vertex is the point where the parabola
changes direction.
The figure below shows the graph of the parabola.
The standard form of the equation of a parabola with vertex (h, k) and axis parallel to a
coordinate axis is:
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
or
𝑥 = 𝑎(𝑦 − ℎ)2 + 𝑘
Parabola Centered at (h, k)
Let’s have an example. Write the equation 𝑦 = (3𝑥)2 − 12𝑥 + 16 in the form
𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
𝑦 = (3𝑥)2 − 12𝑥 + 16
𝑦 = 3(𝑥 2 − 4𝑥) + 16
𝑦 = 3(𝑥 2 − 4𝑥 + 4𝑥) + 16 − 12
𝒚 = 𝟑(𝒙 − 𝟐)𝟐 + 𝟒
Factor 3𝑥 2 − 12𝑥
Complete the square
Graph of Parabola
The parabola has the following characteristics: vertex, axis of symmetry, focus, directrix, and opening.
Let us graph the parabola with equation 𝑦 = 2𝑥 2 + 4𝑥 + 5
The parabola in standard form can be transformed in vertex form.
Compare how the graph of a parabola makes a shift. What is your observation?
It’s your turn. Transform the standard form in vertex form.
Module 1 in Pre-Calculus
Name: ____________________________
Yr/Sec: ___________________________
Teacher: _______________________
Score: _________________________
Written Work
Exercise 1
A. Direction: Classify each equation as that of a circle, a parabola, an ellipse, a hyperbola, a
degenerate conic, or none of these.
1
3
4
2
(𝑥 − 2)2 + (𝑦 + 3)2 = 16
(𝑥 − 2)2 + (𝑦 + 3)2 = 16
(𝑥 − 3)2 (𝑦 − 4)2
+
=1
4
9
𝑥 2 /49 − 𝑦 2 /16 = 1
5
𝑥 3 + 𝑦 4 + 5𝑥 2 = 1
B. Direction: Rearrange the letters to form a Mathematical word. Provide a short meaning of
description for each.
1. IN COCS 2. CER LICS 3. DEER ATE GEN 4. L PERA BOY H 5. LARO PA BA 6. SPEL LIE -
C. Instruction: Now that you have the center and radius for each equation, sketch its graph. Label
the circle according to the number.
1. (𝑦 − 3)2 + (𝑥 − 1)2 = 9
4. (𝑥)2 + (𝑦)2 − 6𝑥 = 7
2. (𝑦 − 1)2 + (𝑥 − 5)2 = 25
5. (𝑥)2 + (𝑦)2 − 14𝑥 + 2𝑦 = −14
3. (𝑦 + 7)2 + (𝑥 + 5)2 = 49
Performance Tasks
(Module 1)
A. Write each equation in the form 𝑦 = 𝑎(𝑥 − ℎ)2 + 𝑘
1. 𝑦 = 5𝑥 2 − 10𝑥 + 8
3. 𝑦 = −𝑥 2 + 10𝑥 − 15
2. 𝑦 = 2𝑥 2 − 8𝑥 + 2
B. Give the features of the graph of equation in LETTER A (item 1 and 3)
a. Vertex:
b. Axis of symmetry:
c. Focus:
d. Directrix:
e. Opening:
“LOVE is a higher frequency than fear; therefore we must overcome
fear before reaching LOVE”
Mr. Jon – Jon C. Marcelino
Teacher