Topics in
Analytic Geometry
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10
10.2
INTRODUCTION TO CONICS: PARABOLAS
Conics
A conic section (or simply conic) is the intersection of a
plane and a double-napped cone. Notice below, that in the
formation of the four basic conics, the intersecting plane
does not pass through the vertex of the cone.
Circle
Ellipse
Parabola
Hyperbola
Basic Conics
Figure 10.9
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Conics
When the plane does pass through the vertex, the resulting
figure is a degenerate conic, as shown in Figure 10.10.
Point
Line
Two Intersecting Lines
Degenerate Conics
Figure 10.10
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Conics
There are several ways to approach the study of conics.
You could begin by defining conics in terms of the
intersections of planes and cones, as the Greeks did, or
you could define them algebraically, in terms of the general
second-degree equation
Ax2 + Bxy + Cy2 + Dx + Ey + F = 0.
We will study a third approach, in which each of the conics
is defined as a locus (collection) of points satisfying a
geometric property.
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x2  y 2  r 2
Conics
For example, a circle is defined as the collection of all points (x, y) that
are equidistant from a fixed point (h, k).
This leads to the standard form of the equation of a circle
 x  3   y  2 
2
(x –
h)2
+ (y –
k)2
=r
2.
2
 25
Equation of circle
Center  3, 2 ; r  5
 3,3
5
 2,  2
 3,  2 8,  2
 3,  7 
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Parabolas
We have learned that the graph of the quadratic function
f(x) = ax2 + bx + c
is a parabola that opens upward or downward.
The following definition of a parabola is more general in the
sense that it is independent of the orientation of the
parabola.
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Parabolas
The midpoint between the focus and the directrix is called
the vertex, and the line passing through the focus and the
vertex is called the axis of the parabola.
A parabola is symmetric with respect to its axis.
Parabola
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Parabolas
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Parabolas
10
Parabolas
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Parabolas
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Example 1 – Vertex at the Origin
Find the standard equation of the parabola with vertex at
the origin and focus (2, 0).
The axis of the parabola is horizontal, passing through
(0, 0) and (2, 0).
The standard form is y2 = 4px,
where h = 0, k = 0, and p = 2. So,
the equation is y2 = 8x.
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Find the standard form of the equation of the parabola
with the given characteristics and vertex at the orgin.
20)
y  4 px  36  4 p  2 
2
36
9
p 
8
2
2
 y  1 8 x
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Find the standard form of the equation of the parabola
with the given characteristics and vertex at the orgin.
22)
 3 
Focus:   , 0 
 2 
x
 3 
F   ,0
 2 
3
2
V  0,0 
3
y  4 px  p  
2
 y 2  6 x
2
15
Find the standard form of the equation of the parabola
with the given characteristics and vertex at the orgin.
26)
Directrix: y  2
x 2  4 py  p  2
 x2  8 y
F  0, 2
V  0,0 
y  2
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Find the standard form of the equation of the parabola
with the given characteristics and vertex at the orgin.
30) Vertical Axis and passes through  3, 3
y 2  4 px  9  4 p  3 
3
p
4
 y 2  3 x
x
3
4
 3 
F   ,0
 4 
 3, 3
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Find the vertex, focus, and directrix of the
parabola, and sketch its graph.
34) y  2 x 2
1
x  4 py  4 p  
2
1
 p   and V  0, 0 
8
2
y
1
8
1

F  0,  
8

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Find the vertex, focus, and directrix of the
parabola, and sketch its graph.
2
42)  x  1   4  y  1


2

 x  h
2
 4p y  k
 1 
 p  1 and V   ,1
 2 
 1 
F   ,2
 2 
 1 
V   ,1
 2 
y0
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Find the standard form of the equation of the parabola
with the given characteristics.
52)
 y  k   4 p  x  h
2
 4  3  4 p  4.5  5 
y  4 px
2
2
1  2 p
1
p
2
  y  3  2  x  5 
2
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Application
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Example 4 – Finding the Tangent Line at a Point on a Parabola
Find the equation of the tangent line to the parabola given
by y = x2 at the point (1, 1).
2
x  4 py
For this parabola,
and the focus is
.
9 5
 1 
16 4
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Example 4 – Solution
cont’d
Setting d1 = d2 produces
So, the slope of the tangent line is
 0, 1
y = 2x – 1.
HW p. 738/19-31 EOO, 33-45 EOO, 51-59 EOO
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