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EQUATIONS OF CONICS MATCHED WITH DEFINITIONS OF CONIC
SECTIONS
1. Write the equation of the given ellipse.
2. Use a few points on the ellipse to verify that the sum of the
distances from any point on the ellipse to the two
foci is constant. What is this constant sum for the given
ellipse?
3. Write the equation of the given circle.
4. Use a few points on the circle to verify that the distance
from any point on the circle to the center is constant.
What is this constant distance for the given circle?
6
4
2
2
4
6
10 5 5 10(10, 0)(-10, 0) (8, 0)(-8, 0)
6
4
2
2
4
6
10 5 5 10
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5. Write the equation of the given hyperbola.
6. Use a few points on the hyperbola to verify that the
difference of the distances from any point on the
hyperbola to the two foci is constant. What is this constant
difference for the given hyperbola?
7. Write the equation of the given parabola.
8. Use a few points on the parabola to verify that the distance
from any point on the parabola to the focus is
equal to the distance from that point to the directrix.
6
4
2
2
4
6
10 5 5 10(5, 0)(-5, 0)(-8, 0) (8, 0)
6
4
2
2
4
6
10 5 5 10
y = -3
(0, 3)
(0, 0)
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CONICS EQUATIONS
b2=a
2-c
2
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b2=c
2-a
2
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Geometric and Parametric Equations of Conic Sections
Shape
Geometric Equation
Center at (0, 0)
Geometric Equation
Center at (h, k)
Parametric Equation
Center at (0, 0)
Parametric Equation
Center at (h, k)
ELLIPSE
with longer part a
along x-axis
x2
a2y
2
b2 1
(x h)2
a2
(y k)2
b2 1
x = a cos (t)
y = b sin (t)
x = a cos (t) + h
y = b sin (t) + k
ELLIPSE
with longer part a
along y-axis
x2
b2y
2
a2 1
(x h)2
b2
(y k)2
a2 1
x = b sin (t)
y = a cos (t)
x = b sin (t) + h
y = a cos (t) + k
CIRCLE
x2
r 2y
2
r2 1
OR
x2 y
2 r
2
(x h)2
r 2
(y k)2
r 2 1
OR
(x h)2 (y k)
2 r
2
x = r cos (t)
y = r sin (t)
x = r cos (t) + h
y = r sin (t) + k
HYPERBOLA
with curves opening
left and right
x2
a2y
2
b21
(x h)2
a2
(y k)2
b21
x = a / cos (t)
y = b tan (t)
x = a / cos (t) + h
y = b tan (t) + k
HYPERBOLA
with curves opening
up and down
y2
a2x
2
b21
(y k)2
a2
(x h)2
b21
x = b tan (t)
y = a / cos (t)
x = b tan (t) + h
y = a / cos (t) + k
PARABOLA
opening up or down
4p y = x2
OR
y 1
4px
2
4p (y - k) = (x - h)2
OR
y 1
4p(x h)
2 k
x = t
y 1
4pt
2
x = t + h
y 1
4pt
2 + k
PARABOLA
opening left or right
y2 = 4p x
OR
y 4px
(y k)2 4p(x h)
OR
y 4p(x h) k
x 1
4pt
2
y = t
x 1
4pt
2 + h
y = t + k
Note: can be expressed as by completing the square.
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STANDARD FORMS OF CONIC EQUATIONS A common form for the equation
of any conic section with major and minor axes parallel to the x-
and y-axes is Ax2 + Cy2 + Dx + Ey + F = 0. What must be true of the
general conic equation Ax2 + Cy2 + Dx + Ey + F = 0 in order for the
graph of the equation to be
(a) an ellipse? (b) a hyperbola? (c) a parabola?
Graph and compare the following conics: ELLIPSES HYPERBOLAS
PARABOLAS 1. x2 + 3y2 56 = 0 2. 3x2 + y2 56 = 0 3. x2 + 3y2 8x +
12y 28 = 0 4. 3x2 + y2 24x + 4y 4 = 0
5. 25x2 - 5y2 88 = 0 6. - 5x2 + 25y2 88 = 0 7. 25x2 5y2 + 120x +
36y 8.8 = 0 8. - 5x2 + 25y2 24x 180y + 207.2 = 0
9. 4x2 + 6(2) y = 0 10. 4y2 + 6(2) x = 0 11.
12.
Next, consider the addition of the term B x y to the form Ax2 +
Cy2 + Dx + Ey + F = 0. What must be true of the general conic
equation Ax2 + B x y + Cy2 + Dx + Ey + F = 0 in order for the graph
of the equation to be
(a) an ellipse? (b) a hyperbola? (c) a parabola?
How does the B x y term affect the shape of a conic? In other
words, what does the graph of an equation in Ax2 + B x y + Cy2 + Dx
+ Ey + F = 0 form look like? How is it different from the graph of
Ax2 + Cy2 + Dx + Ey + F = 0 ? ELLIPSES HYPERBOLAS PARABOLAS 13. x2
+ xy + y2 28 = 0 14. x2 + xy + y2 6x + 0y 16 = 0
15. x2 + 3xy + y2 8.8 = 0 16. x2 + 3xy + y2 6x + 0y 16 = 0
17. x2 + 2xy + y2 3x + 3y = 0 18. x2 + 2xy + y2 6x + 0y 16 =
0
4x2+55
3x + 6 2y +
3025
144!91 2
4
"
#$%
&'= 0
4y2+ 6 2x !
91
3y +
8281
144+55 2
4
"
#$%
&'= 0
