1. Lesson 10.3-10.4

2.  Slicing these cones with a plane at different
angles produces different conic sections.

3.  For example, you can describe a circle as a
locus of points that are a fixed distance from
a fixed point.
 Definition of a Circle
◦ A circle is a locus of points P in a plane, that are a
constant distance, r, from a fixed point, C.
Symbolically, PC r. The fixed point is called the
center and the constant distance is called the
radius.

5. EX 1 Write an equation of a circle with
center (3, -2) and a radius of 4.
2 2 2
+x h y k r
22 2
3 + 2 4x y
2 2
3 + 2 16x y

6. EX 2 Write an equation of a circle with
center (-4, 0) and a diameter of 10.
2 2 2
+x h y k r
2 2 2
4 + 0 5x y
2 2
4 +y 25x

7. EX 4 Find the coordinates of the center and
the measure of the radius.
2 2 2
6 + 3 25x y

8. 1. Move the x terms together and the
y terms together.
2. Move C to the other side.
3. Complete the square (as needed)
for x.
4. Complete the square(as needed)
for y.
5. Factor the left & simplify the right.

9. 2 2
4 6 3x x y y
2 2
4 6 3 0x y x y
Center: (-2, 3) radius: 4
2 2
2 3 16x y
2 2
4 6 9394 4x x y y

10. Find the equation of the circle whose endpoints of a diameter
are (11, 18) and (-13, -20):
Center is the midpoint of the diameter
11 13 18 20
, 1 1,
2 2
Radius uses distance formula
2 2
1 2 1 2r x x y y
2 2
r 11 1 18 1
r 505
2 2
r 13 1 20 1
r 505
22 2
1 1 05x 5y

12. Write an equation in standard form of an ellipse that has a vertex at
(0, –4), a co-vertex at (3, 0), and is centered at the origin.
Since (0, –4) is a vertex of the ellipse, the other vertex is at (0, 4), and
the major axis is vertical.
Since (3, 0) is a co-vertex, the other co-vertex is at (–3, 0), and the
minor axis is horizontal.
So, a = 4, b = 3, a2 = 16, and b2 = 9.
+ = 1 Standard form for an equation of an
ellipse with a vertical major axis.
(x-h) 2
b2
(y-k) 2
a2
+ = 1 Substitute 9 for b2 and 16 for a2.
(x-0) 2
9
(y-0) 2
16
An equation of the ellipse is + = 1.
x 2
9
y 2
16

13.  b) Find coordinates of vertices,
covertices, foci
 Center = (-3,2)
 Horizontal ellipse since the a²
value is under x terms
 Since a = 3 and b = 2
 Vertices are 3 points left and
right from center  (-3 3, 2)
 Covertices are 2 points up and
down  (-3, 2 2)
 Now to find focus points
 Use c² = a² - b²
 So c² = 9 – 4 = 5
 c² = 5 and c = √5
 Focus points are √5 left and
right from the center  F(-3
√5 , 2)
1
4
)2y(
9
)3x( 22 • a) GRAPH
• Plot Center (-3,2)
• a = 3 (go left and right)
• b = 2 (go up and down)

14. Find the foci of the ellipse with the equation 9x2 + y2 = 36. Graph the
ellipse.
9x2 + y2 = 36
Since 36 > 4 and 36 is with y2, the major axis is vertical, a2 = 36, and b2 = 4.
+ = 1 Write in standard form.
x 2
4
y 2
36
c2 = a2 – b2 Find c.
= 36 – 4 Substitute 4 for a2 and 36 for b2.
= 32
The major axis is vertical, so the coordinates of the foci are (0, c).
The foci are: (0, 4 2 ) and (0, – 4 2).
c = 32 = 4 2