chapter 9 analytic geometry. section 9-1 distance and midpoint formulas
TRANSCRIPT
Chapter 9
Analytic Geometry
Section 9-1Distance and
Midpoint Formulas
Pythagorean Theorem
If the length of the hypotenuse of a right triangle is c, and the lengths of the other two sides are a and b, then c2 = a2 + b2
Example 8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
F
D
E
Find the distance between point D and point F.
Distance Formula
D = √(x2 – x1)2 + (y2 – y1)2
Example
Find the distance between points A(4, -2) and B(7, 2)
d = 5
Midpoint Formula
M( x1 + x2, y1 + y2)
2 2
Example
Find the midpoint of the segment joining the points (4, -6) and (-3, 2)
M(1/2, -2)
Section 9-2
Circles
Conics
Are obtained by slicing a double cone
Circles, Ellipses, Parabolas, and Hyperbolas
Equation of a Circle
The circle with center (h,k) and radius r has
the equation
(x – h)2 + (y – k)2 = r2
Example
Find an equation of the circle with center (-2,5) and radius 3.
(x + 2)2 + (y – 5)2 = 9
Translation
Sliding a graph to a new position in the coordinate plane without changing its shape
Translation
8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
Example
Graph (x – 2)2 + (y + 6)2 = 4
6
4
2
-2
-4
-6
-8
-10
-10 -5 5 10
Example
If the graph of the equation is a circle, find its center and radius.
x2 + y2 + 10x – 4y + 21 = 0
Section 9-3
Parabolas
Parabola
A set of all points equidistant from a fixed line called the directrix, and a fixed point not on the line, called the focus
Vertex
The midpoint between the focus and the directrix.
Parabola - Equations
y-k = a(x-h)2
Vertex (h,k) symmetry x = h
x - h = a(y-k)2
Vertex (h,k) symmetry y = k
Equation of a Parabola
Remember:
y – k = a(x – h)2
(h,k) is the vertex of the parabola
Example 1
The vertex of a parabola is (-5, 1) and the directrix is the line y = -2. Find the focus of the parabola.
(-5 4)
Example 1 8
6
4
2
-2
-4
-6
-8
-10 -5 5 10
directrix (y = -2)
f x = -2
Vertex (-5,1)
Example 2
Find an equation of the parabola having the point F(0, -2) as the focus and the line x = 3 as the directrix.
y – k = a(x – h)2
a) a = 1/4c where c is the distance between the vertex and focus
b) Parabola opens upward if a>0, and downward if a< 0
y – k = a(x – h)2
c) Vertex (h, k)
d) Focus (h, k+c)
e) Directrix y = k – c
f) Axis of Symmetry x = h
x - h = a(y –k)2
a) a = 1/4c where c is the distance between the vertex and focus
b) Parabola opens to the right if a>0, and to the left if a< 0
x – h = a(y – k)2
c) Vertex (h, k)
d) Focus (h + c, k)
e) Directrix x = h - c
f) Axis of Symmetry y = k
Example 3
Find the vertex, focus, directrix , and axis of symmetry of the parabola:
y2 – 12x -2y + 25 = 0
Example 4
Find an equation of the parabola that has vertex (4,2) and directrix y = 5
Section 9-4
Ellipses
Ellipse
The set of all points P in the plane such that the sum of the distances from P to two fixed points is a given constant.
Focus (foci)
Each fixed point Labeled as F1 and F2
PF1 and PF2 are the focal radii of P
Ellipse- major x-axis
drag
Ellipse- major y-axis
drag
Example 1
Find the equation of an ellipse having foci (-4, 0) and (4, 0) and sum of focal radii 10. Use the distance formula.
Example 1 - continued
Set up the equation
PF1 + PF2 = 10
√(x + 4)2 + y2 + √(x – 4)2 + y2 = 10Simplify to get x2 + y2 = 1
25 9
Graphing
The graph has 4 intercepts
(5, 0), (-5, 0), (0, 3) and (0, -3)
Symmetry
The ellipse is symmetric about the x-axis if the denominator of x2 is larger and is symmetric about the y-axis if the denominator of y2 is larger
Center
The midpoint of the line segment joining its foci
General Form
x2 + y2 = 1 a2 b2
The center is (0,0) and the foci are (-c, 0) and (c, 0) where
b2 = a2 – c2
focal radii = 2a
General Form
x2 + y2 = 1 b2 a2
The center is (0,0) and the foci are (0, -c) and (0, c) where
b2 = a2 – c2
focal radii = 2a
Finding the Foci
If you have the equation, you can find the foci by solving the equation b2 =a2 – c2
Example 2
Graph the ellipse
4x2 + y2 = 64
and find its foci
Example 3
Find an equation of an ellipse having x-intercepts √2 and - √2 and y-intercepts 3 and -3.
Example 4
Find an equation of an ellipse having foci (-3,0) and (3,0) and sum of focal radii equal to 12.
Section 9-5
Hyperbolas
Hyperbola
The set of all points P in the plane such that the difference between the distances from P to two fixed points is a given constant.
Focal (foci)
Each fixed point Labeled as F1 and F2
PF1 and PF2 are the focal radii of P
Example 1
Find the equation of the hyperbola having foci (-5, 0) and (5, 0) and difference of focal radii 6. Use the distance formula.
Example 1 - continued
Set up the equation
PF1 - PF2 = ± 6
√(x + 5)2 + y2 - √(x – 5)2 + y2 = ± 6Simplify to get x2 - y2 = 1
9 16
Graphing
The graph has two x-intercepts and no y-intercepts
(3, 0), (-3, 0)
Asymptote(s)
Line(s) or curve(s) that approach a given curve arbitrarily, closely
Useful guides in drawing hyperbolas
Center
Midpoint of the line segment joining its foci
General Form
x2 - y2 = 1 a2 b2
The center is (0,0) and the foci are (-c, 0) and (c, 0), and difference of focal radii 2a where b2 = c2 – a2
Asymptote Equations
y = b/a(x) and
y = - b/a(x)
General Form
y2 - x2 = 1 a2 b2
The center is (0,0) and the foci are (0, -c) and (0, c), and difference of focal radii 2a where b2 = c2 – a2
Asymptote Equations
y = a/b(x)
and
y = - a/b(x)
Example 2
Find the equation of the hyperbola having foci (3, 0) and (-3, 0) and difference of focal radii 4. Use the distance formula.
Example 3
Find an equation of the hyperbola with asymptotes
y = 3/4x and y = -3/4x and foci (5,0) and (-5,0)
Section 9-6More on Central Conics
Ellipses with Center (h,k)
• Horizontal major axis: (x –h)2 + (y-k)2 = 1
a2 b2
Foci at (h-c,k) and (h + c,k) where c2 = a2 - b2
Ellipses with Center (h,k)
• Vertical major axis: (x –h)2 + (y-k)2 = 1 b2 a2
Foci at (h, k-c) and (h,c +k) where c2 = a2 - b2
Hyperbolas with Center (h,k)
• Horizontal major axis: (x –h)2 - (y-k)2 = 1
a2 b2
Foci at (h-c,k) and (h + c,k) where c2 = a2 + b2
Hyperbolas with Center (h,k)
• Vertical major axis: (y –k)2 - (x-h)2 = 1 a2 b2
Foci at (h, k-c) and (h, k+c) where c2 = a2 + b2
Example 1
Find an equation of the ellipse having foci (-3,4) and (9, 4) and sum of focal radii 14.
Example 2
Find an equation of the hyperbola having foci
(-3,-2) and (-3, 8) and difference of focal radii 8.
Example 3
Identify the conic and find its center and foci, graph.
x2 – 4y2 – 2x – 16y – 11 = 0