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
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.
3
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).
4
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).
5
(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.
6
Write an equation of the ellipse with foci at (3,
5) and (3, 1) and vertices at (3, 6) and (3,
2).
SOLUTION
Plot the given points and make a rough sketch.
7
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.
8
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 b 2 a 2 c 2 .
9
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.
10
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.
11
Here are all of the conic section equations in
standard form

• parabola y a(x h)2 k x
  a(y k)2 h

circle (x h)2 (y k)2 r2
hyperbola (x h)2 (y k)2 1
a2 b2 (y k)2 (x h)2 1
a2 b2
ellipse (x h)2 (y k)2 1 a2
b2
12
How do you write an equation in standard form?
Example Write the equation of the ellipse in
standard form 2x2 3y2 4x 12y 10 0
Group the xs and ys together...
2x2 4x 3y2 12y 10
Factor out the GCFs...
Complete the square for each variable.
2(x2 2x ) 3(y2 4y ) 10
1 4
2 12
What will make each a perfect square trinomial?
Add the real amount to the other side (remember
that they are being distributed)
Rewrite as the squares of binomials...
2(x 1)2 3(y 2)2 24
Divide to set the right side equal to 1...
2(x 1)2 3(y 2)2 24 24 24
24
(x 1)2 (y 2)2 1 12 8
13
Lets try anotherwith graphing.
Example Graph the hyperbola 9x2 4y2 18x
16y 43 0
Group the xs and ys together...
9x2 18x 4y2 16y 43
Factor out the GCFs...
Complete the squares ...
9(x2 2x ) 4(y2 4y ) 43
1 4
9 16
9(x 1)2 4(y 2)2 36
9(x 1)2 4(y 2)2 36 36 36
36
(x 1)2 (y 2)2 1 4 9
Center (1, 2)
a 2 b 3