This document discusses conic sections and circles. It defines conic sections as curves formed by the intersection of a plane and a double right circular cone. The main conic sections are parabolas, ellipses, hyperbolas, and circles. Circles are specifically defined as sets of points equidistant from a fixed center point. The document derives the standard and general forms of the equation of a circle, and provides examples of writing equations of circles given properties like center and radius.

1. CONIC
SECTIONS
Jay-Art F. Agustin
TSU – LABORATORY SCHOOL Pre-Calculus for Grade 11 STEM

2. Objectives:
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● Illustrate the different types of conic sections – parabola,
hyperbola, ellipse, circle, and degenerate cases;
● Define a circle;
● Determine the standard from of equation of a circle; and
● Graph a circle in the rectangular coordinate plane.
TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

3. Conic Sections
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

4. The Origin
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
Apollonius of Perga
● Greek Geometer
● Studies the curves formed
by the intersection of a
plane and a double right
circular cone.

5. The ‘Cone-ic’ Sections
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry
● Conic section is a curve
formed by the
intersection of a plane
and a double right circular
cone.

6. The ‘Cone-ic’ Sections
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

7. The Three Conic Sections
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● Parabola is formed if the cutting plane is parallel to one and
only one generator
● Ellipse is formed if the cutting of the plane is not parallel to
any generator.
○ Circle is formed if the cutting of the plane is not parallel
to any to any generator but is perpendicular to the axis.
● Hyperbola is formed if the cutting plane is parallel to two
generators.
TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

8. The Three Conic Sections
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

9. Degenerate Conic Sections
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

10. Conic Sections as Defined
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● A conic 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.
TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

11. Elements
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Focus (F) – the fixed point of a conic
Directrix (d) – the fixed line d
corresponding to the focus.
Principal axis (a) – a line that
passes through the focus and
perpendicular to the directrix. Every
conic is symmetric with respect to its
principal axis.
TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

12. Elements
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Vertex (V) – 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 |FP| to the distance |QP|, which
is a constant. In symbols,
𝒆 =
|𝑭𝑷|
|𝑸𝑷|
TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

13. Eccentricity
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● The conic is a parabola if the
eccentricity 𝑒 = 1.
● The conic is an ellipse if the
eccentricity 𝑒 < 1.
● The conic is circle if the
eccentricity 𝑒 = 0.
● The conic is a hyperbola if the
eccentricity 𝑒 > 1.
TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

14. Circles
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TSU – LABORATORY SCHOOL M106: Calculus I with Analytic Geometry

15. Circle Defined
A circle 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.
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16. Circle Defined
A circle 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.
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17. Equation of a Circle with C (0, 0)
𝑥2
+ 𝑦2
= 𝑟2
This equation is referred to as the
standard form of equation of a circle
whose center is at the origin with
radius r.
This can be derived using the
distance formula.
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18. Derivation
𝑥2
+ 𝑦2
= 𝑟2
The distance of C (0,0) to P (x, y) is equal to the radius.
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19. Equation of a Circle with C (h,k)
(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2
This equation of the circle whose
center is at the point (ℎ, 𝑘) and with
radius 𝑟
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20. Exercise
Determine the standard form of equation of the following circles
and the radius. Draw the graph.
1. Center 𝐶(0, 0), radius: 9;
2. Center 𝐶(−5,7), radius: 6;
3. Center 𝐶( 7, 3 3), radius: 8.
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21. Exercise
1. Center 𝐶(0, 0), radius: 9
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22. Exercise
1. Center 𝐶(−5,7), radius: 6
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23. Exercise
1. Center 𝐶( 7, 3 3), radius: 8
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24. Equation of a Circle in GF
𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎,
where 𝐷 = −2ℎ, E = −2k, and F = ℎ2
+ 𝑘2
− 𝑟2
This general form was derived by expanding the standard form of
the equation.
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25. Derivation
𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎,
where 𝐷 = −2ℎ, E = −2k, and F = ℎ2
+ 𝑘2
− 𝑟2
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(𝑥 − ℎ)2
+(𝑦 − 𝑘)2
= 𝑟2

26. Examples
Write the equation of a circle in general form with center at (−3, −2) and
radius of 4.
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27. Examples
Write the equation of a circle in general form with center at (−4, 6) and
radius of 5.
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28. Examples
Write the equation of a circle in general form with center at (2, −4) and
radius of 5.
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29. Examples
Determine the center and radius of the circle in general form.
𝑥2 + 𝑦2 − 4𝑥 − 2𝑦 − 4 = 0
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30. Examples
Determine the center and radius of the circle in general form.
𝑥2 + 𝑦2 − 10𝑥 − 6𝑦 − 18 = 0
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31. Examples
Determine the center and radius of the circle in general form.
2𝑥2 + 2𝑦2 − 16𝑥 + 12𝑦 − 4 = 0
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32. Equation of a Circle in GF
𝒙𝟐 + 𝒚𝟐 + 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎,
where 𝐷 = −2ℎ, E = −2k, and F = ℎ2
+ 𝑘2
− 𝑟2
Solving for r in terms of D, E, and F in the general form:
If
𝐷2
4
+
𝐸2
4
− 𝐹 > 0, then the graph of the equation is a circle.
If
𝐷2
4
+
𝐸2
4
− 𝐹 = 0, then the graph of the equation is a point circle.
If
𝐷2
4
+
𝐸2
4
− 𝐹 < 0, the equation has no graph.
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33. Examples
Determine whether the equation represents a circle, a point circle, or
has no graph.
𝑥2 + 𝑦2 + 8𝑥 + 15 = 0
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34. Examples
Determine whether the equation represents a circle, a point circle, or
has no graph.
𝑥2 + 𝑦2 + 87𝑥 + 5𝑦 + 16 = 0
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35. Examples
Determine whether the equation represents a circle, a point circle, or
has no graph.
𝑥2 + 𝑦2 + 2𝑥 − 6𝑦 + 12 = 0
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36. Examples
Find the general equation of a circle.
The center of the circle is at ( -3, 7) and goes through the origin.
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37. Examples
Find the general equation of a circle.
The center of the circle is at (7, 4) and goes through the point (4, 8)
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38. Examples
Find the equation of a circle
The circle passes through the origin, and contain the points (0, 5), and (3, 3).
𝑥2
+ 𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
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39. Examples
Find the equation of a circle
The circle passes through the origin, and contain the points (0, 8), and (5, 5).
𝑥2
+ 𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
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40. Examples
Find the equation of a circle
The circle passes through the points (2, 3), (6, 1), and (4, -3).
𝑥2
+ 𝑦2
+ 𝐷𝑥 + 𝐸𝑦 + 𝐹 = 0
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41. Examples
A particular cellphone tower is designed to service a 15-km radius. The tower is located at
(5, -3) on a coordinate plane whose units represents km.
● What is the standard form equation of the outer boundary of the region serviced by
the tower?
● What is the general form equation of the region serviced by the tower?
● Draw the graph of the region serviced by the tower.
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42. Meronnnn paaaa!
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