This document discusses conic sections and circles. It defines conic sections as sections obtained when a plane cuts through a circular cone. It defines a circle as the set of all points equidistant from a fixed center point, where the fixed distance is called the radius. The document also mentions that Apollonius, a Greek mathematician, studied conic sections and gave them their names. He believed they should be studied for their mathematical beauty rather than practical applications.

1. PRE-CALCULUS
GRADE 11 – FIRST SEMESTER
Conic Sections: Circle

2. CONIC SECTIONS
• or simply CONICS are simply the sections
obtained if a plane is made to cut a right
circular cone
• defined as the path of a point which moves so
that its distance from a fixed point (FOCUS) is
in a constant ratio (ECCENTRICITY) to its
distance from a fixed line (DIRECTRIX)

3. APOLLONIUS
• Greek mathematician who wrote the CONIC
SECTIONS
• He gave the names of the conics and believed
that they should be studied for the beauty of
demonstrations rather than for practical
applications

4. Determine the quadrant or axis where
a given point can be located.
1. 𝑨 (𝟒, −𝟓)
2. 𝑹 (𝟎, 𝟗)
3. 𝑽 (−𝟖, −𝟒. 𝟓)
4. 𝑰 ( −
𝟏𝟒
𝟑
, 𝟎)
5. 𝑵 (−𝟕. 𝟖, 𝟓. 𝟓𝟑)

5. Plot the following points in a
rectangular coordinate plane.
1. J (9, -14)
2. O (
𝟑
𝟒
,
𝟏
𝟐
)
3. N1 (-7, -6.5)
4. N2 ( −
𝟏𝟕
𝟑
, 𝟔. 𝟔)
5. A (-4.8, 9.44)

6. Get a sheet of paper and follow the
steps below:
1. Draw a point somewhere in the middle part
of your sheet of paper.
2. Now, using a ruler, mark 20 other points that
are 5 cm from the first point.
3. Compare your work with that of your
seatmates.
4. What shape do you recognize?

8. Activity.
Your grandfather told you that when he was
young, he and his playmates buried some old
coins under the ground, thinking that these
coins will be valuable after several years. He also
remembered that these coins were buried
exactly 4 kilometres from Tree A (see map) and 5
kilometres from Tree B. Where could the coins
possibly be located?

9. Map

10. DEFINITION OF CIRCLE
• CIRCLE is defined as the set of all points whose
distance from a fixed point called center is
constant. The fixed distance from the center
to any point on the circle is called RADIUS.

11. DERIVATION OF THE EQUATION
OF A CIRCLE

12. 𝒙𝟐
+ 𝒚𝟐
= 𝒓𝟐
or
𝒙 − 𝒉 𝟐
+ 𝒚 − 𝒌 𝟐
= 𝒓𝟐
−𝑠𝑡𝑎𝑛𝑑𝑎𝑟𝑑 𝑓𝑜𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒 𝑜𝑟
𝑡ℎ𝑒 𝑐𝑒𝑛𝑡𝑒𝑟 − 𝑟𝑎𝑑𝑖𝑢𝑠 𝑓𝑜𝑟𝑚
𝑨𝒙𝟐
+ 𝑨𝒚𝟐
+ 𝑫𝒙 + 𝑬𝒚 + 𝑭 = 𝟎
𝑔𝑒𝑛𝑒𝑟𝑎𝑙 𝑓𝑜𝑟𝑚 𝑜𝑓 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑜𝑓 𝑎 𝑐𝑖𝑟𝑐𝑙𝑒
𝑎2
+ 𝑏2
= 𝑐2
𝑃𝑦𝑡ℎ𝑎𝑔𝑜𝑟𝑒𝑎𝑛 𝑡ℎ𝑒𝑜𝑟𝑒𝑚
𝑥2 + 𝑦2 = 𝑟2

13. Illustrative Examples
1. Find the equation of a circle whose point is at
(4, 6) and center at the origin. Sketch the
graph.
Solution: 𝑥2
+ 𝑦2
= 𝑟2
𝑟 = 42 + 62 = 16 + 36 = 52
𝑥2
+ 𝑦2
= 52
2
𝑥2
+ 𝑦2
= 52

15. 2. Find the general equation of a circle with
circle at point (-3, 5) with radius equal to 4.
Sketch the graph.
Solution: 𝒙 − 𝒉 𝟐
+ 𝒚 − 𝒌 𝟐
= 𝒓𝟐
𝑥 − (−3) 2
+ 𝑦 − 5 2
= 42
𝑥 + 3 2
+ 𝑦 − 5 2
=16
𝑥2
+ 6𝑥 + 9 + 𝑦2
− 10𝑦 + 25 = 16
Answer:
𝑥2
+ 𝑦2
+ 6𝑥 − 10𝑦 + 18 =0

17. 3. Reduce the equation 𝑥2
+ 𝑦2
+ 4𝑥 − 6𝑦 − 12 =0
to standard form and find the coordinates of the
center and radius. Sketch the graph.
Solution: 𝑥2
+ 𝑦2
+ 4𝑥 − 6𝑦 − 12 =0
𝑥2
+ 4𝑥 + 4 + 𝑦2
− 6𝑦 + 9 = 12 + 4 + 9
𝑥 + 2 2
+ 𝑦 − 3 2
=25
Answer: 𝑥 + 2 2
+ 𝑦 − 3 2
=52
𝐶 (−2, 3), 𝑟 = 5

19. GAME (CUBING)
Face 2.
Center (2,-5) and
radius 6
Face 3.
Center (-3,6) and
radius 4
Face 6.
Center (-1,7) and
radius 3
Face 1.
Center (1,4) and
radius 8
Face 4.
Center (-4,1) and
radius 7
Face 5.
Center (-4,-5) and
radius 5
Each group will roll a die where
each face have a given centre and
radius. Write the equation in both
standard and general form and
graph the circle in a coordinate
plane.

20. EXERCISES
Reduce the following equations to standard
form and find the center and radius of the circle.
1. 𝑥2
+ 𝑦2
− 4𝑥 + 6𝑦 + 12 = 0
2. 𝑥2
+ 𝑦2
− 8𝑥 + 2𝑦 − 16 = 0
3. 5𝑥2
+ 5𝑦2
+ 10𝑥 − 5𝑦 + 3 = 0
4. 3𝑥2
+ 3𝑦2
− 2𝑥 + 𝑦 − 5 = 0

21. Real-life Problems Involving Circles
1. OCEAN NAVIGATION The beam of a lighthouse can be seen for up to 20 miles.
You are on a ship that is 10 miles east and 16 miles north of the lighthouse.
a. Write an inequality to describe the region lit by the lighthouse beam.
b. Can you see the lighthouse beam?
SOLUTION
a. As shown at the right the lighthouse beam can be
seen from all points that satisfy this inequality:
𝒙𝟐 + 𝒚𝟐 < 𝟐𝟎𝟐
b. Substitute the coordinates of the ship into the
inequality you wrote in part (a).
𝒙𝟐
+ 𝒚𝟐
< 𝟐𝟎𝟐
Inequality from part (a)
𝟏𝟎2 + 𝟏𝟔2 <? 202Substitute for x and y.
100 + 256 <? 400 Simplify.
356 < 400 The inequality is true.
You can see the beam from the ship.

22. 2. OCEAN NAVIGATION Your ship in Example 1 is traveling due south.
For how many more miles will you be able to see the beam?
SOLUTION:
When the ship exits the region lit by the beam,
it will be at a point on the circle 𝑥2 + 𝑦2 = 202.
Furthermore, its x-coordinate will be 10 and its
y-coordinate will be negative. Find the point
(𝟏𝟎, 𝒚) where 𝑦 < 0 on the circle 𝑥2
+ 𝑦2
= 202
.
𝑥2
+ 𝑦2
= 202
. Equation for the boundary
102 + 𝑦2 = 202 Substitute 10 for x.
𝒚 = ± 300 ≈ ±𝟏𝟕. 𝟑 Solve for y.
Since 𝑦 < 0, 𝑦 ≈ −17.3. The beam will be in view as the ship travels
from (10, 16) to (10, o17.3), a distance of |16 − (−17.3)| =
33.3 𝑚𝑖𝑙𝑒𝑠.

23. 3. RADIO SIGNALS The signals of a radio station can be received
up to 65 miles away. Your house is 35 miles east and 56 miles
south of the radio station. Can you receive the radio station’s
signals? Explain.
Solution:
the radio station’s signals can be received from all points that
satisfy this inequality:
𝒙𝟐 + 𝒚𝟐 < 𝟔𝟓𝟐
352
+ (−56)2
<? 652
1225 + 3136 <? 4225
4361 < 4225 the inequality is FALSE.
Thus, the signal will not reach the house.

25. Test
1. Define conics/ conic sections. (3 points)
2. Define circle (2 points)
3. Define radius ( 2 points)
4. He was the Greek mathematician who gave
the names of the conics. (1 point)
5. What was his belief about conics? (1 point)
6. Write the standard form of the equation of a
circle if the center is at the origin.
7. Write the general form of the equation of a
circle.

26. A. Write the standard form of the equation of the
circle with the given radius and whose centre is at
the origin.
8. 3 10. 5 6
9. 7
B. Write the standard form of the equation of the
circle that passes through the given point and
whose centre is at the origin.
11. 0, −10 12. 5, −3
C. 13.Reduce 𝑥2
+ 𝑦2
+ 5𝑥 − 16𝑦 − 10 =0 in
center-radius form. Give the coordinates of the
center and the length of the radius.

27. ASSIGNMENT
Find any circular object (e.g. mouth of a circular
glass, ring) in your home. Measure the radius in
terms of centimetre or inch. Write an equation
for that circular object assuming its center is at
the origin.