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Sec= ond level
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Conic Sections
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Conic Sections
Conic sections are the cross sections of right circular cones.
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Conic Sections
Conic sections are the cross sections of right circular cones. 
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
 • hyperbolas
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
 • hyperbolas
Where as = straight lines are the graphs of 1st degree equations
Ax +  By =3D C, conic sec= tions are the = graphs of 2nd degree equations in x and y.
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
 • hyperbolas
Where as = straight lines are the graphs of 1st degree equations
Ax +  By =3D C, conic sec= tions are the = graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax= 2 + By2 + Cx + Dy =3D E, where A, B, C, D,= and E are numbers.
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
 • hyperbolas
Where as = straight lines are the graphs of 1st degree equations
Ax +  By =3D C, conic sec= tions are the = graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax= 2 + By2 + Cx + Dy =3D E, where A, B, C, D,= and E are numbers. We are to match these 2nd degree equations with = the diffe= rent conic sections.
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
 • hyperbolas
Where as = straight lines are the graphs of 1st degree equations
Ax +  By =3D C, conic sec= tions are the = graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax= 2 + By2 + Cx + Dy =3D E, where A, B, C, D,= and E are numbers. We are to match these 2nd degree equations with the diffe= rent conic sections. The algebraic technique that enable us= to sort out which equation corresponds to which conic section is called "completing the square".
A right circular cone
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Conic Sections
Conic sections are the= cross sections of right circular cones.  <= /span>There are= four different types of curves:
 • circles&#= 13;
 • ellipses
 • parabolas
 • hyperbolas
Where as = straight lines are the graphs of 1st degree equations
Ax +  By =3D C, conic sec= tions are the = graphs of 2nd degree equations in x and y. In particular, the conic sections that are parallel to the axes (not tilted) have equations of the form Ax= 2 + By2 + Cx + Dy =3D E, where A, B, C, D,= and E are numbers. We are to match these 2nd degree equations with the diffe= rent conic sections. The algebraic technique that enable us= to sort out which equation corresponds to which conic section is called "completing the square".
We start = with the Distance Formula.
A right circular cone
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
Conic Sections
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1) 
Conic Sections
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx 
Conic Sections
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Conic Sections
Δy =3D the difference between the y's =3D = y2 – y1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Conic Sections
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Example A.  Find the <= /span>= distance between (2, –1) and (–2, 2).=
Conic Sections
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Example A.  Find the = distance between (2, –1) and (–2, 2).
Δy =3D (–1) – (2) = =3D –3
Δy=3D-3
=
Conic Sections
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Example A.  Find the = distance between (2, –1) and (–2, 2).
Δy =3D (–1) – (2) = =3D –3
Δx =3D (2) –= (–2) =3D 4
Δy=3D-3
Δx=3D4
Conic Sections
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Example A.  Find the = distance between (2, –1) and (–2, 2).
Δy =3D (–1) – (2) = =3D –3
Δx =3D (2) –= (–2) =3D 4r =3D Ö(–3)2 + 42 =3D Ö25 =3D 5
Δy=3D-3
Δx=3D4
r=3D5
Conic Sections
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Example A.  Find the = distance between (2, –1) and (–2, 2).
Δy =3D (–1) – (2) = =3D –3
Δx =3D (2) –= (–2) =3D 4r =3D Ö(–3)2 + 42 =3D Ö25 =3D 5
Δy=3D-3
Δx=3D4
r=3D5
Conic Sections
The geometric definiti= on of all four types of conic sections are
distance relations bet= ween points.
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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The Distance Formula:&= #13;
Given two points P =3D= (x1, y1) and Q = =3D (x2, y2) in the xy-plane, the distance r between P and Q is:
r =3D Ö(y– y= 1)2 + (x2 – x1)  =3D ÖΔy2 +Δx where
Example A.  Find the = distance between (2, –1) and (–2, 2).
Δy =3D (–1) – (2) = =3D –3
Δx =3D (2) –= (–2) =3D 4r =3D Ö(–3)2 + 42 =3D Ö25 =3D 5
Δy=3D-3
Δx=3D4
r=3D5
Conic Sections
The geometric definiti= on of all four types of conic sections are
distance relations bet= ween points.  We start with the ci= rcles.
Δy =3D the difference between the y's =3D = y2 – y1
Δx =3D the difference between the x's =3D = x2 – x1
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Circles
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r<= /div>
r<= /div>
Circles
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
C
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r<= /div>
r<= /div>
The radiu= s and the center completely determine the circle.
Circles
center
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r
The radiu= s and the center completely determine the circle.
Circles
Let (h, k) be the = center of a circle and r be the radius.
(h, k)
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r
The radiu= s and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the = center of a circle and r be the radius.  Suppose (x, y) is a po= int on the circle, then the distance between (x, y) and the center is r.
(h, k)
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r
The radiu= s and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the = center of a circle and r be the radius.  Suppose (x, y) is a po= int on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r= =3D Ö (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2
        = ;     
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r
The radiu= s and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the = center of a circle and r be the radius.  Suppose (x, y) is a po= int on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r= =3D Ö (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2
        = ;       or
r= 2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
 
 
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r
The radiu= s and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the = center of a circle and r be the radius.  Suppose (x, y) is a po= int on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r= =3D Ö (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2
        = ;       or
r= 2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
 
 
This is called the standard = form <= /span>of circles.
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r
The radiu= s and the center completely determine the circle.
Circles
(x, y)
Let (h, k) be the = center of a circle and r be the radius.  Suppose (x, y) is a po= int on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r= =3D Ö (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2
        = ;       or
r= 2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
 
 
This is called the standard form of circles. Given an equation of this f= orm, we can easily identify the center and the radius.
A circle is the set of= all the points that have equal distance r, called the radius, to a fi= xed point C which is called the center.
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r2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
Circles
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r2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
must be &= #8220; – ”
Circles
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r2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
r is the r= adius
must be &= #8220; – ”
Circles
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r2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
r is the r= adius
must be &= #8220; – ”
(<= span lang=3DEN-US style=3D'font-size:133%;color:red;mso-fareast-language:ZH-CN'= >h, k) is the = center
Circles
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r2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 
r is the r= adius
must be &= #8220; – ”
(<= span lang=3DEN-US style=3D'font-size:133%;color:red;mso-fareast-language:ZH-CN'= >h, k) is the = center
Circles
Example B.  Write the equation <= /span>
of the c= ircle as shown.
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r2 =3D (x – h)<= span lang=3DEN-US style=3D'font-size:89%;position:relative;top:-.45em;mso-text-= raise: 30%;mso-fareast-language:ZH-CN'>2 + (y – k