Download Conics The conics get their name from the fact that they can be

Conics
The conics get their name from the fact that they can be formed by passing a plane through a double-napped
cone. There are four conic sections. The general form of a second degree equation is given by Ax2 + Bxy + Cy2
+ Dx + Ey + F = 0. Checking the discriminant, we can determine the type of conic we are dealing with.
b2 – 4ac < 0 ; B = 0 and A = C it is a circle
b2 – 4ac < 0 and either B ≠ 0 or A ≠ C it is an ellipse
b2 – 4ac = 0 it is a parabola
b2 – 4ac > 0 it is a hyperbola
Determining Conic Sections by Inspection
To determine the conic section by inspection, complete any squares that are necessary, so that the variables are
on one side and the constant is on the right hand side. Any squared variable below could be replaced by a
quantity. That is, instead of x2 + y2 = 1, it might be (x - 2)2 + y2 = 1
Circle
x2 + y2 = 1
Both squared terms are present, both are positive, both have the same coefficient.
Ellipse
3x2 + 4y2 = 1 or
Both squared terms are present, both are positive, but they have different coefficients.
Hyperbola
x2 - y2 = 1
Both squared terms are present, but one is positive and the other is negative. The coefficients may or
may not be the same, it doesn't matter.
Parabola
x2 + y = 1
Both variables are present, but one is squared and the other is linear.
Line
x+y=1
Neither variable is squared.
Point
x2 + y2 = 0
A circle (or ellipse) with the right hand side being zero.
No Graph
x2 + y2 = -1
A circle (or ellipse) with the right hand side being negative.
Intersecting Lines
x2 - y2 = 0
A hyperbola with the right hand side equal to zero.
Parallel Lines
x2 = 1
One variable is squared and the other variable is missing. The right hand side must be positive. If the
right hand side is zero, then it is a line (x2 = 0 so x = 0) and if the right hand side is negative (x2 = -1),
then there is no graph.
Parabola
A parabola is "the set of all points in a plane equidistant from a fixed point (focus) and a fixed line (directrix)".
The distances to any point (x,y) on the parabola from the focus (0,p) and the directrix y=-p, are equal to each
other. This can be used to develop the equation of a parabola. The standard form is x2 = 4py.
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The starting point is the vertex at (h,k)
There is an axis of symmetry that contains the focus and the vertex and is perpendicular to the directrix.
Move p units along the axis of symmetry from the vertex to the focus.
Move -p units along the axis of symmetry from the vertex to the directrix (which is a line).
The focus is within the curve.
Standard Form of Equations for Parabolas
Form
(x – h)2 = 4p(y – k)
(y – k)2 = 4p(x – h)
Orientation
opens vertically
opens horizontally
Vertex
(h, k)
(h, k)
Focus
(h, k + p)
(h + p, k)
Axis of Symmetry
x=h
y=k
Directrix d:
y=k–p
x=h–p
The line tangent to a parabola forms an isosceles triangle where the segment from a point P to the focus forms
one leg of the triangle and the segment along the axis of symmetry from the focus to another point on the
tangent line forms the other leg.
The parabola has the property that any signal (light, sound, etc) entering the parabola parallel to the axis of
symmetry will be reflected through the focus (this is why satellite dishes and those parabolic antennas that the
detectives use to eavesdrop on conversations work). Also, any signal originating at the focus will be reflected
out parallel to the axis of symmetry (this is why flashlights work).
Circle
A circle is "the set of all points in a plane equidistant from a fixed point (center)".
The standard form for a circle, with center at the origin is x2 + y2 = r2, where r is the radius of the circle. The
formula is (x – h)2 + (y – k)2 = r2 when the center is (h, k).
Ellipse
An ellipse is "the set of all points in a plane such that the sum of the distances from two fixed points (foci) is
constant".
The sum of the distances to any point on the ellipse (x,y) from the two foci (c,0) and (-c,0) is a constant. That
constant will be 2a.
If we let d1 and d2 be the distances from the foci to the point, then d1 + d2 = 2a.
The ellipse is a stretched circle. Begin with the unit circle (circle with a radius of 1) centered at the origin.
Stretch the vertex from x=1 to x=a and the point y=1 to y=b. What you have done is multiplied every x by a and
multiplied every y by b.
In translation form, you represent that by x divided by a and y divided by b. So, the equation of the circle
changes from x2 + y2 = 1 to (x/a)2 + (y/b)2 = 1 and that is the standard equation for an ellipse centered at the
origin.
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The center is the starting point at (h,k).
The major axis contains the foci and the vertices.
Major axis length = 2a. This is also the constant that the sum of the distances must add to be.
Minor axis length = 2b.
Distance between foci = 2c.
The foci are within the curve.
Since the vertices are the farthest away from the center, a is the largest of the three lengths, and the
Pythagorean relationship is: a2 = b2 + c2.
Standard Form of Equations for Ellipses
Form
( x  h) 2 ( y  k ) 2
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1
a2
b2
( x  h) 2 ( y  k ) 2

1
b2
a2
Orientation
horizontal major axis
vertical major axis
Center
(h, k)
(h, k)
Foci
(h ± c, k)
(h, k ± c)
Vertices
(h ± a, k)
(h, k ± a)
Co-vertices
(h, k ± b)
(h ± b, k)
Major Axis
y=k
x=h
Minor Axis
x= h
y=k
a, b, c relationship:
c2 = a2 – b2
c2 = a2 – b2
c
which is the ratio of c to a. This value will always be between 0 and 1 and
a
will determine how “circular” or “stretched” the ellipse will be.
eccentricity
e
Hyperbola
A hyperbola is "the set of all points in a plane such that the difference of the distances from two fixed points
(foci) is constant".
The difference of the distances to any point on the hyperbola (x,y) from the two foci (c,0) and (-c,0) is a
constant. That constant will be 2a.
If we let d1 and d2 be the distances from the foci to the point, then | d1 - d2 | = 2a.
The absolute value is around the difference so that it is always positive.
The only difference in the definition of a hyperbola and that of an ellipse is that the hyperbola is the difference
of the distances from the foci that is constant and the ellipse is the sum of the distances from the foci that is
constant.
Instead of the equation being (x/a)2 + (y/b)2 = 1, the equation is (x/a)2 - (y/b)2 = 1.
The graphs, however, are very different.
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The center is the starting point at (h,k).
The Transverse axis contains the foci and the vertices.
Transverse axis length = 2a. This is also the constant that the difference of the distances must be.
Conjugate axis length = 2b.
Distance between foci = 2c.
The foci are within the curve.
Since the foci are the farthest away from the center, c is the largest of the three lengths, and the
Pythagorean relationship is: a2 + b2 = c2.
Standard Form of Equations for Hyperbolas
Form
( x  h) 2 ( y  k ) 2

1
a2
b2
( y  k ) 2 ( x  h) 2

1
a2
b2
Orientation
horizontal tranverse axis
vertical transverse axis
Vertices
(h ± a, k)
(h, k ± a)
Foci
(h ± c, k)
(h, k ± c)
Transverse Axis
y=k
x=h
Conjugate Axis
x= h
y=k
Asymptotes
b
y  k   ( x  h)
a
a
y  k   ( x  h)
b
a, b, c relationship:
c2 = a2 + b2
c2 = a2 + b2
c
which is the ratio of c to a. This value will always be greater than 1 and
a
will determine how “stretched” the hyperbola will be.
eccentricity
e