32 conic sections, circles and completing the square
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Conic Sections
Conic SectionsOne way to study a solid is to slice it open.
Conic SectionsOne way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area.
Conic Sections
A right circular cone
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A Horizontal Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A Horizontal Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A Moderately Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A Moderately Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A Horizontal Section
A Moderately Tilted Section
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Circles and ellipsis are enclosed.
Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
A Parallel–Section
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
A Parallel–Section
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
An Cut-away Section
Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
An Cut-away Section
Conic Sections
A right circular cone and conic sections (wikipedia “Conic Sections”)
An Cut-away Section
One way to study a solid is to slice it open. The exposed area of the sliced solid is called a cross sectional area. Conic sections are the borders of the cross sectional areas of a right circular cone as shown.
Parabolas and hyperbolas are open.
A Horizontal Section
A Moderately Tilted Section
Circles and ellipsis are enclosed.
A Parallel–Section
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Conic Sections
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Conic Sections
Besides their differences in visual appearance and the manners they reside inside the cone, there are many reasons, that have nothing to do with cones, that the conic sections are grouped into four groups.
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Conic Sections
Besides their differences in visual appearance and the manners they reside inside the cone, there are many reasons, that have nothing to do with cones, that the conic sections are grouped into four groups. One way is to use distance relations to classify them.
We summarize the four types of conics sections here.
Circles Ellipses
Parabolas Hyperbolas
Conic Sections
Besides their differences in visual appearance and the manners they reside inside the cone, there are many reasons, that have nothing to do with cones, that the conic sections are grouped into four groups. One way is to use distance relations to classify them. We use the circles and the ellipsis as examples.
CirclesGiven a fixed point C,
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r.
C
r
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r.
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r.
r
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r.
r
C
r
Hence a dog tied to a post would mark offa circular track.
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
r
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
F2F1
Given two fixed points (called foci),
r
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
r
C
r
F2F1
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
F2F1
PFor example, if P, Q, and R are points on a ellipse,
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
Q
R
r
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
F2F1
P
p1
p2
For example, if P, Q, and R are points on a ellipse, thenp1 + p2
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
Q
R
r
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
F2F1
P
p1
p2
For example, if P, Q, and R are points on a ellipse, thenp1 + p2
= q1 + q2
q1
q2
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
Q
R
r
C
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
F2F1
P
p1
p2
For example, if P, Q, and R are points on a ellipse, thenp1 + p2
= q1 + q2
= r1 + r2
= a constant
q1
q2
r2r1
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
Q
R
r
C
r
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
C
F2F1
p1
p2
For example, if P, Q, and R are points on a ellipse, thenp1 + p2
= q1 + q2
= r1 + r2
= a constant
q1
q2
r2r1
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
r
Q
R
P
Hence a dog leashed by a ring to two posts would mark offan elliptical track.
r
CirclesGiven a fixed point C, a circle is the set of points whose distances to C is a fixed constant r. The equal-distance r is called the radius and the point C is called the center of the circle.
C
F2F1
p1
p2
For example, if P, Q, and R are points on a ellipse, thenp1 + p2
= q1 + q2
= r1 + r2
= a constant
q1
q2
r2r1
Given two fixed points (called foci), an ellipse is the set of points whose sum of the distances to the foci is a constant.
r
Q
R
P
Likewise parabolas and hyperbolas may be defined using relations of distance measurements.
Conic SectionsThe second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
Conic Sections
Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
Conic Sections
Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
y = –1 y + x = 1 x = 1
Linear graphs
Conic Sections
Conic sections are the graphs of 2nd degree equations in x and y.
Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
y = –1 y + x = 1 x = 1
Linear graphs
Conic Sections
Conic sections 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 formAx2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.
Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
y = –1 y + x = 1 x = 1
Linear graphs
Conic Sections
Conic sections 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 formAx2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.
Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
y = –1 y + x = 1 x = 1
The algebraic technique that enables us to sort these 2nd degree equations into four groups of conic sections is called "completing the square".
Linear graphs
Conic Sections
Conic sections 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 formAx2 + By2 + Cx + Dy = E, where A, B, C, D, and E are numbers.
Recall that straight lines are the graphs of 1st degree equations Ax + By = C where A, B, C, are numbers.
The second reason that we group the conic sections into four types is algebraic, i.e. the equations related to graphs of the conic sections can easily be sorted into the above four types
y = –1 y + x = 1 x = 1
The algebraic technique that enables us to sort these 2nd degree equations into four groups of conic sections is called "completing the square". We will apply this method to the circles but only summarize the results about the other ones.
Linear graphs
rr
Circles
center
A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
rr
The radius 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 fixed point C which is called the center.
r
The radius 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 fixed point C which is called the center.
r
The radius 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 point 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 fixed point C which is called the center.
r
The radius 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 point on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r = (x – h)2 + (y – k)2
A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
r
The radius 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 point on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r = (x – h)2 + (y – k)2
orr2 = (x – h)2 + (y – k)2
A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
r
The radius 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 point on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r = (x – h)2 + (y – k)2
orr2 = (x – h)2 + (y – k)2 This is called the standard form of circles.
A circle is the set of all the points that have equal distance r, called the radius, to a fixed point C which is called the center.
r
The radius 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 point on the circle, then the distance between (x, y) and the center is r. Hence,
(h, k)
r = (x – h)2 + (y – k)2
orr2 = (x – h)2 + (y – k)2 This is called the standard form of circles. Given an equation of this form, 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 fixed point C which is called the center.
r2 = (x – h)2 + (y – k)2
Circles
r2 = (x – h)2 + (y – k)2
must be “ – ”Circles
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”Circles
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example A. Write the equation of the circle as shown.
(–1, 3)
(–1, 8)
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example A. Write the equation of the circle as shown.
The center is (–1, 3) and the radius is 5.
(–1, 3)
(–1, 8)
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example A. Write the equation of the circle as shown.
The center is (–1, 3) and the radius is 5. Hence the equation is:52 = (x – (–1))2 + (y – 3)2
(–1, 3)
(–1, 8)
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example A. Write the equation of the circle as shown.
The center is (–1, 3) and the radius is 5. Hence the equation is:52 = (x – (–1))2 + (y – 3)2 or25 = (x + 1)2 + (y – 3 )2
(–1, 3)
(–1, 8)
r2 = (x – h)2 + (y – k)2
r is the radius must be “ – ”
(h, k) is the center
Circles
Example A. Write the equation of the circle as shown.
The center is (–1, 3) and the radius is 5. Hence the equation is:52 = (x – (–1))2 + (y – 3)2 or25 = (x + 1)2 + (y – 3 )2
(–1, 3)
(–1, 8)
In particular a circle centered at the origin has an equation of the form x2 + y2 = r2
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Circles
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the standard form:
42 = (x – 3)2 + (y – (–2))2
Circles
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the standard form:
42 = (x – 3)2 + (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the standard form:
42 = (x – 3)2 + (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
(3, 2)
(3, --2)
(3, --6)
(7, --2) (--1, --2)
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the standard form:
42 = (x – 3)2 + (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square".
(3, 2)
(3, --2)
(3, --6)
(7, --2) (--1, --2)
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the standard form:
42 = (x – 3)2 + (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square". To complete the square means to add a number to an expression so the sum is a perfect square.
(3, 2)
(3, --2)
(3, --6)
(7, --2) (--1, --2)
Example B. Identify the center and the radius of 16 = (x – 3)2 + (y + 2)2. Label the top, bottom, left and right most points. Graph it.
Put 16 = (x – 3)2 + (y + 2)2 into the standard form:
42 = (x – 3)2 + (y – (–2))2
Hence r = 4, center = (3, –2)
Circles
When equations are not in the standard form, we have to rearrange them into the standard form. We do this by "completing the square". To complete the square means to add a number to an expression so the sum is a perfect square. This procedure is the main technique in dealing with 2nd degree equations.
(3, 2)
(3, --2)
(3, --6)
(7, --2) (--1, --2)
The Completing the Square MethodCircles
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square,
Circles
CirclesThe Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd degree equation into the standard form.
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd degree equation into the standard form.1. Group the x2 and the x-terms together, group the y2 and y terms together, and move the number term to the other side of the equation.
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
The Completing the Square MethodIf we are given x2 + bx, then adding (b/2)2 to the expression makes the expression a perfect square, i.e. x2 + bx + (b/2)2 is the perfect square (x + b/2)2.
Circles
Example C. Fill in the blank to make a perfect square.
a. x2 – 6x + (–6/2)2 = x2 – 6x + 9 = (x – 3)2
b. y2 + 12y + (12/2)2 = y2 + 12y + 36 = ( y + 6)2
The following are the steps in putting a 2nd degree equation into the standard form.1. Group the x2 and the x-terms together, group the y2 and y terms together, and move the number term to the other side of the equation. 2. Complete the square for the x-terms and for the y-terms. Make sure to add the necessary numbers to both sides.
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3, –6),and radius is 3.
Circles
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3, –6),and radius is 3.
Circles
(3, –6),
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3, –6),and radius is 3.
Circles
(3, –6),(6, –6),
(3, –3),
(0, –6),
(–9, –6)
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3, –6),and radius is 3.
Circles
(3, –6),(6, –6),
(3, –3),
(0, –6),
(–9, –6)
The Completing-the-Square method is the basic method for handling 2nd degree problems.
Example E. Use completing the square to find the center and radius of x2 – 6x + y2 + 12y = –36. Find the top, bottom, left and right most points. Graph it.
We use completing the square to put the equation into the standard form:x2 – 6x + + y2 + 12y + = –36 complete the squares x2 – 6x + 9 + y2 + 12y + 36 = –36 + 9 + 36 ( x – 3 )2 + (y + 6)2 = 9 ( x – 3 )2 + (y + 6)2 = 32 Hence the center is (3, –6),and radius is 3.
Circles
(3, –6),(6, –6),
(3, –3),
(0, –6),
(–9, –6)
The Completing-the-Square method is the basic method for handling 2nd degree problems.We summarize the hyperbola and parabola below.
Hyperbolas
HyperbolasJust as all the other conic sections, hyperbolas are defined by distance relations.
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
Just as all the other conic sections, hyperbolas are defined by distance relations.
A
If A, B and C are points on a hyperbola as shown
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
B
C
Just as all the other conic sections, hyperbolas are defined by distance relations.
A
a2
a1
If A, B and C are points on a hyperbola as shown then a1 – a2
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
B
C
Just as all the other conic sections, hyperbolas are defined by distance relations.
A
a2
a1
b2
b1
If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
B
C
Just as all the other conic sections, hyperbolas are defined by distance relations.
A
a2
a1
b2
b1
If A, B and C are points on a hyperbola as shown then a1 – a2 = b1 – b2 = c2 – c1 = constant.
c1
c2
Hyperbolas
Given two fixed points, called foci, a hyperbola is the set of points whose difference of the distances to the foci is a constant.
B
C
Just as all the other conic sections, hyperbolas are defined by distance relations.
Parabolas
Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below.For more information, see: http://en.wikipedia.org/wiki/Parabola
Finally, we illustrate the definition that’s based on distance measurements of the parabolas.
F
L
Parabolas
Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below.For more information, see: http://en.wikipedia.org/wiki/Parabola
Finally, we illustrate the definition that’s based on distance measurements of the parabolas.
F
L
a
A
P1
Parabolas
Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below.For more information, see: http://en.wikipedia.org/wiki/Parabola
Finally, we illustrate the definition that’s based on distance measurements of the parabolas.
F
L
ab
A
B
P1
P2
Parabolas
Given a fixed point F, and a line L, the points that are of equal distance from F the line L is a parabola. Hence a = A, b = B, c = C as shown below.For more information, see: http://en.wikipedia.org/wiki/Parabola
Finally, we illustrate the definition that’s based on distance measurements of the parabolas.
ab
AcB
C
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