Reviewer
Subject: Precalculus
Exam: Midterm Examination
Topic Focus: Conic Sections (Circle, Parabola, Ellipse)
1. Introduction to Conic Sections
Conic sections are curves formed when a plane intersects a double-
napped cone.
The type of curve depends on the angle of the plane relative to the
cone’s axis:
o Circle → plane cuts perpendicular to axis (one focus).
o Ellipse → plane cuts at an angle but not steep enough to
intersect both nappes (two foci).
o Parabola → plane is parallel to a generating line of the cone
(focus & directrix).
o Hyperbola → plane cuts both nappes of the cone (two branches,
eccentricity > 1).
Degenerate conics occur when the plane passes through the
vertex of the cone.
o Point, line, or pair of intersecting lines
2. The Circle
Definition:
A circle is the set of all points in a plane equidistant from a fixed point
(the center).
Distance from center to any point is called the radius.
Standard Equation:
( x−h )2+ ( y−k )2=r 2
Center: (h , k )
Radius: r
Examples:
1. Center (0 , 0), radius 5 → x 2+ y 2=25.
2. Center (3 ,−2), radius 5 → ( x−3 )2 + ( y +2 )2=25.
Applications:
Parks, gardens, wheels, coins all model circular equations.
3. The Parabola
Definition:
A parabola is the set of all points equidistant from a fixed point
(focus) and a fixed line (directrix).
The midpoint between focus and directrix is the vertex.
Standard Equations:
Opens Up/Down: x 2=4 py or x 2=4 a y
Opens Left/Right: y 2=4 px or x 2=4 a y
Vertex: (0 , 0), Focus: (0 , p), Directrix: y=− p .
Key Properties:
Axis of symmetry: line through vertex and focus.
Distance between vertex and focus = distance between vertex and
directrix.
Example:
Focus (0,3), vertex (0,0) → equation: x 2=12 y .
Opens upward because focus is above directrix.
Applications:
Parabolic mirrors, satellite dishes, fountains, projectiles.
4. The Ellipse
Definition:
An ellipse is the set of all points where the sum of distances from
two fixed points (foci) is constant.
Standard Equations:
Horizontal major axis:
( x−h )2 ( y −k )2
+ =1 , a>b
a2 b2
Vertical major axis:
( x−h )2 ( y −k )2
+ =1 , a>b
b2 a2
Key Properties:
a=semi−major a xis
b=semi−minor axis
c=distance ¿ center ¿ foci .
Relationship: c 2=a2−b 2 .
Examples:
1. Ellipse centered at (0,0), vertices at (±5,0), co-vertices (0,±3):
2 2
x y
+ =1
25 9
2. If center at (–2,1), vertical major axis → denominator under y−term is
larger.
Applications:
Planetary orbits (Earth around the Sun).
Elliptical ponds, stadium tracks.
5. Comparisons Between Conics
Circle: one focus, radius constant.
Ellipse: two foci, constant sum of distances.
Parabola: one focus + one directrix, distance equal.
Hyperbola: two foci, constant difference of distances, eccentricity > 1.
6. Tips for Solving Exam Problems
Identify the conic: look at the equation form.
Check orientation:
o Circle: equal denominators (or coefficients).
o Ellipse: larger denominator tells orientation (under x 2 →
horizontal, under y 2 → vertical).
o Parabola: look at squared term (if x 2 → opens up/down, if y 2 →
opens left/right).
Find center, radius, vertices, foci using standard forms.
Sketch by plotting center, axis lengths, and orientation.
7. Sample Problem Review
1. Equation of a Circle: Find the equation of a circle with center (–2,4)
and radius 6.
( x +2 )2+ ( y−4 )2=36
2. Equation of a Parabola: Find the parabola with vertex (0,0) and
focus (0,–3).
o Opens downward → x 2=−12 y .
3. Equation of an Ellipse: Center (0,0), major axis along y-axis, vertices
at (0 , ± 6), foci(0 , ± 5).
2 2
x y
+ =1
11 36
