2. LEARNING OBJECTIVES
At the end of this lesson, you are expected to:
• illustrate the different types of conic sections: parabola,
ellipse, circle, hyperbola, and degenerate cases
• define a circle
• determine the standard form of equation of a circle
• define a parabola
• determine the standard form of equation of a parabola
14. INTERACTIVE 3D OF CONIC
SECTIONS
https://www.intmath.com/plane-analytic-
geometry/conic-sections-summary-interactive.php
15. COMMON PARTS OF THE CONIC SECTIONS
The conic sections may have looked different; however,
they still have common parts.
VERTEX
CENTER
FOCUS
DIRECTRIX
16. COMMON PARTS OF THE CONIC SECTIONS
VERTEX
Vertex is an extreme point on a parabola, hyperbola, and
ellipse. Although, ellipse has vertices and co-vertices.
with horizontal axis
17. COMMON PARTS OF THE CONIC SECTIONS
VERTEX
Vertex is an extreme point on a parabola, hyperbola, and
ellipse. Although, ellipse has vertices and co-vertices.
with vertical axis
18. COMMON PARTS OF THE CONIC SECTIONS
FOCUS AND DIRECTRIX
The focus and directrix are the point and the line on a conic section
that are used to define and construct the curve respectively. The
distance of any point on the curve from the focus to the directrix is
proportion as shown on the images below by the green lines. In a
plane, the circle has no defined directrix.
with horizontal axis
19. COMMON PARTS OF THE CONIC SECTIONS
FOCUS AND DIRECTRIX
The focus and directrix are the point and the line on a conic section
that are used to define and construct the curve respectively. The
distance of any point on the curve from the focus to the directrix is
proportion as shown on the images below by the green lines. In a
plane, the circle has no defined directrix.
with vertical axis
20. COMMON PARTS OF THE CONIC SECTIONS
CENTER
For circles, the center is the point equidistant from any
point on the surface.
21.
22.
23.
24.
25.
26. Looking at the Cartesian plane, the
vertex is at the point (0,0) since it is
the extreme point of the parabola.
The focus is at (3,0).
In order to solve for the directrix,
we need to look at the orientation of
the parabola. Since the formula has
a horizontal axis, the formula for the
directrix will be 𝑥 = −𝑐. Thus, the
equation of the directrix is 𝑥 = −3.
27.
28. LET’S TRY
Given the curve on the Cartesian plane below, identify the
focus, vertex, and directrix.
29.
30. Looking at the graph, we can see
that the foci are at (−2 − 3) and (6,
−3).
Note that the center is the midpoint
of the foci. Thus, we have the
following solution:
34. Step 1: Plot the vertex and the
focus, and identify the curve.
Looking at the Gateway Arch, it
looks like a parabola.
Step 2: Solve for the directrix.
Since the focus is at (0, −3),
the directrix is 𝑦 = 3
44. A. Identify the conic section or the part that is being
described.
1. These are the conic sections that are formed when the
plane intersects the double-napped cone in such a way that
it passes through the apex.
2. This conic section is formed when the plane is parallel to
the axis of revolution.
3. It is the midpoint of the two foci for ellipse and hyperbola.
4. It refers to the extreme point of a parabola.
5. These are the curves that are obtained between the
intersection of a double-napped cone and a plane.
ASSESSMENT
45. B.Using the image below, complete the table and
solve for the directrix given the vertices and foci.
46.
47.
48. Assignment: Challenge Yourself
Answer the following questions:
1. You saw Albert playing with a double-napped cone and a paper. He
put the paper on top of one cone and said that he was able to form a
conic section. Do you agree with him? Explain your answer.
2. A glass was placed on the table. If you hold a flashlight as shown
below, what kind of curve will be formed by its shadow?
3. If you shift a parabola with vertex at the origin, two units to the right,
what will be the new vertex?