In the following equations the point h, k is the vertex of the parabola and the center of the other

1
In the following equations the point (h, k) is
the vertex of the parabola and the center of the
other conics.
(x h) 2 (y k) 2 r 2
CIRCLE
2
CIRCLE
(x h) 2 (y k) 2 r 2
3
PARABOLA
(y k) 2 4p (x h)
Opens left or right
(x h) 2 4p (y k)
Opens up or down
4
Horizontal Ellipse
Vertical Ellipse
5
Hyperbolas
Opens left and right
Opens up and down
6
In the following equations the point (h, k) is
the vertex of the parabola and the center of the
other conics.
(x h) 2 (y k) 2 r 2
CIRCLE
7
Write an equation of the parabola whose vertex is
at (2, 1) and whose focus is at (3, 1).
SOLUTION
Find h and k The vertex is at (2, 1), so h
2 and k 1.
(y 1) 2 4p(x 2).
8
Write an equation of the parabola whose vertex is
at (2, 1) and whose focus is at (3, 1).
SOLUTION
(2, 1)
The standard form of the equation is (y 1) 2
4(x 2).
9
Graph (x 3) 2 (y 2) 2 16.
SOLUTION
Compare the given equation to the standard form
of the equation of a circle
(3, 2)
(x h) 2 (y k) 2 r 2
You can see that the graph will be a circle with
center at (h, k) (3, 2).
10
(3, 2)
Graph (x 3) 2 (y 2) 2 16.
SOLUTION
r
The radius is r 4
( 1, 2)
(3, 2)
(7, 2)
Plot several points that are each 4 units from
the center
(3, 6)
(3 4, 2 0) (7, 2)
(3 4, 2 0) ( 1, 2)
(3 0, 2 4) (3, 2)
(3 0, 2 4) (3, 6)
Draw a circle through the points.
11
Write an equation of the ellipse with foci at (3,
5) and (3, 1) and vertices at (3, 6) and (3,
2).
SOLUTION
12
Write an equation of the ellipse with foci at (3,
5) and (3, 1) and vertices at (3, 6) and (3,
2).
SOLUTION
Find a The value of a is the distancebetween
the vertex and the center.
Find c The value of c is the distancebetween
the focus and the center.
13
Write an equation of the ellipse with foci at (3,
5) and (3, 1) and vertices at (3, 6) and (3,
2).
SOLUTION
Find b Substitute the values of a and c into the
equation c 2 a 2 b 2 .
3 2 4 2 b 2
b 2 16 9 7
b 2 7
14
SOLUTION
The y 2-term is positive, so thetransverse axis
is vertical. Sincea 2 1 and b 2 4, you know
thata 1 and b 2.
Plot the center at (h, k) (1, 1). Plot the
vertices 1 unit above and below the center at
(1, 0) and (1, 2).
Draw a rectangle that is centered at (1, 1) and
is 2a 2 units high and 2b 4 units
wide.
15
SOLUTION
The y 2-term is positive, so thetransverse axis
is vertical. Sincea 2 1 and b 2 4, you know
thata 1 and b 2.
Draw the asymptotes through the corners of the
rectangle.
Draw the hyperbola so that it passes through the
vertices and approaches the asymptotes.
16
The equation of any conic can be written in the
form
Ax 2 Bxy Cy 2 Dx Ey F 0
17
Classify the conic 2 x 2 y 2 4 x 4 0.
SOLUTION
Because the coefficients are the same sign and
different, the graph is an ellipse.
18
Classify the conic 4 x 2 9 y 2 32 x 144
y 5 48 0.
SOLUTION
Because the coefficients are opposites,
the graph is a hyperbola.