This document provides instruction on determining the center and radius of a circle given its equation in standard form and vice versa. It begins with stating the objectives of identifying the standard form of a circle equation and using it to determine center and radius or write the equation given one of those. Several examples are worked through, including transforming equations to standard form and finding center and radius. Short exercises are provided for students to practice these skills.
This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
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This powerpoint was used in my 7th and 8th grade classes to review the fundamental counting principle used in our probability unit. There are three independent practice problems at the end.
probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in mathematics 10 .statistics and probability . probability of a mutually and non mutually events. topic in
Circle, Definition, Equation of circle whose center and radius is known, General equation of a circle, Equation of circle passing through three given points, Equation of circle whose diameters is line joining two points (x1, y1) & (x2,y2), Tangent and Normal to a given circle at given point.
Write the equation of a circle given the center and radius
Identify the center and radius of a circle in both center-radius and general form
Write the equation of a circle given the center and a point on the circle
An ellipse is the locus of a point which moves in such a way that its distance form a fixed point is in constant ratio to its distance from a fixed line. The fixed point is called the focus and fixed line is called the directrix and the constant ratio is called the eccentricity of a ellipse denoted by (e).
In other word, we can say an ellipse is the locus of a point which moves in a plane so that the sum of it distances from fixed points is constant.
2.1 Standard Form of the equation of ellipse
Let the distance between two fixed points S and S' be 2ae and let C be the mid point of SS.
Taking CS as x- axis, C as origin.
Let P(h,k) be the moving point Let SP+ SP = 2a (fixed distance) then
(ii) Major & Minor axis : The straight line AA is called major axis and BB is called minor axis. The major and minor axis taken together are called the principal axes and its length will be given by
Length of major axis 2a Length of minor axis 2b
(iii) Centre : The point which bisect each chord of an ellipse is called centre (0,0) denoted by 'C'.
(iv) Directrix : ZM and Z M are two directrix and their equation are x= a/e and x = – a/e.
(v) Focus : S (ae, 0) and S (–ae,0) are two foci of an ellipse.
(vi) Latus Rectum : Such chord which passes through either focus and perpendicular to the major axis is called its latus rectum.
Length of Latus Rectum :
If L is (ae, 𝑙 ) then 2𝑙 is the length of
SP+S'P=
{(h ae)2 k 2} +
= 2a
Latus Rectum.
Length of Latus rectum is given by
2b2
.
h2(1– e2) + k2 = a2(1– e2)
Hence Locus of P(h, k) is given by. x2(1– e2) + y2 = a2(1– e2)
2
a
(vii) Relation between constant a, b, and e
a 2 b2
b2 = a2(1– e2) e2 =
a 2
x2
a 2 +
y
a 2 (1 e 2 ) = 1
e =
a 2
Result :
Major Axis
(a) Centre C is the point of intersection of the axes of an ellipse. Also C is the mid point of AA.
(b) Another form of standard equation of ellipse
x 2 y2
a 2 b2
1 when a < b.
Directrix Minor Axis Directrix x = -a/e x = a/e
Let us assume that a2(1– e2 )= b2
The standard equation will be given by
x2 y2
a2 b2
2.1.1 Various parameter related with standard ellipse :
In this case major axis is BB= 2b which is along y- axis and minor axis is AA= 2a along x- axis. Focus S(0,be) and S(0,–be) and directrix are y = b/e and y = –b/e.
2.2 General equation of the ellipse
The general equation of an ellipse whose focus is (h,k) and the directrix is the line ax + by + c = 0 and the eccentricity will be e. Then let P(x1,y1) be any point on the ellipse which moves such that SP = ePM
Let the equation of the ellipse x
y2
a > b
(x –h)2 + (y –k)2 =
e 2 (ax1 by1 c) 2
a 2 b2
1 1 a 2 b2
(i) Vertices of an ellipse : The point of which ellipse cut the axis x-axis at (a,0) & (– a, 0) and y- axis at (0, b) & (0, – b) is called the vertices of an ellipse.
Hence the locus of (x1,y1) will be given by (a2 + b
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Determining the center and the radius of a circle
1.
2. LEAST MASTERED SKILLS
Determining the Center and the Radius of a Circle
Given its Equation and Vice-Versa
LC CODE: M10GE-IIh-2
OBJECTIVES:
1. Identify the standard form of the equation
of a circle
2. Determine the center and the radius of a
circle given its equation and vice versa
3. Solve real-life problems using equation of a
circle
3. LEAST MASTERED SKILLS
Determining the Center and the Radius of a Circle
Given its Equation and Vice-Versa
LC CODE: M10GE-IIh-2
OBJECTIVES:
1. Identify the standard form of the equation
of a circle
2. Determine the center and the radius of a
circle given its equation and vice versa
3. Solve real-life problems using equation of a
circle
4. As you go through this lesson,
think of this important question:
Perform each activity to find the answer
“How does the equation of a circle facilitates in
finding solutions and making wise decision?”
5. Before you turn the next page,
try to answer the short quiz below...
1. Transform the equation into its standard form.
x2
+y2
+10x+4y-7=0
2. Determine the center and the radius of the following
equation.
1. x2
+ y2
=32
2. x+5)2
+ (y+9)2
=102
3. x2
+y2
+4x-4y-28=0
6. The standard equation of a circle with center at (h,k)
and a radius of r units is (x-h)2 + (y-k)2 =r2 .
7. If the center o the circle is at the
origin, the equation of the circle is
x2 + y2 =r2 .
8. The equation of a circle with center
at (1,3) and radius 5 is
(x-1)2 + (y-3)2 =52
or
(x-1)2 + (y-3)2 =25
9. The equation of a circle with center
at the origin and a radius of 3 is
x2 + y2 =32
or
x2 + y2 =9
10. The equation of a circle with center at
(-5, -9) and radius 10 is
(x+5)2
+ (y+9)2
=102
or
(x+5)2
+ (y+9)2
=100
11. The equation of a circle
with center at the origin
and a radius of 3 is
x2
+ y2
=32
or
x2
+ y2
=9
12. The equation of a circle with
center at
(0, -9) and radius 10 is
x2
+ (y+9)2
=102
or
x2
+ (y+9)2
=100
13. The equation of a circle with center at
(5, 0) and a radius of 4 is
(x-5)2
+ y2
=32
or
(x-5)2
+ y2
=9
14. Suppose two circles have the same center.
Should the equations defining these circles
be the same? Why?
15. The center and the radius of the
circle can be found given the
equation.
To do this, transform the equation
to its standard form. Remember
that the equation will be
(x-h)2
+ (y-k)2
=r2
if the center is
(h, k), or x2
+ y2
=r2
if the center of
the circle is at the origin.
16. Find the center and the radius of the
circle
x2
+ y2
=100.
Solution:
The equation x2
+ y2
=100 has its center
at the origin. Hence it can be trans-
formed to the form
x2
+ y2
= r2
x2
+ y2
= 102
Then the center is at (0, 0) and its radius
is 10.
17. Determine the center and the radius of the
circle (x-5)2
+ (y-8)2
=52
.
The equation (x-5)2
+ (y-8)2
=52
can be written
in the form
(x-h)2
+ (y-k)2
=r2
(x-5)2
+ (y-8)2
=52
(x-5)2
+ (y-8)2
=25
Then the center is at (5, 8) and the radius
is 5.
18. What is the center and the
radius of the circle
x2
+y2
-6x-10y+18=0?
The equation
x2
+y2
-6x-10y+18=0 is written
in general form.
x2
+y2
-6x-10y+18=0
x2
-6x+y2
-10y=-18
Add to both side of the equation:
½(-6)=-3; (-3) 2
=9
and
½(-10)=5; (-5) 2
= 25
Then
x2
-6x+9+y2
-10y+25=-18+9+25
(x2
-6x+9)+(y2
-10y+25)=16
Rewriting, we obtain
(x-3)2
+(y-5)2
=42
Therefore the center is at
(3, 5) and its radius is 4.
19. Write the standard form equation of each of the following circles
given the center and the radius.
Center Radius
1 (3, 8) 1
2 (-6, 4) 3
3 (9, -3) 5
4 (-1, -6) 7
5 (0, 0) 6
6 (0, 5) 4
7 (8, 0) 2
20. Transform the following equation to
standard form, then determine each
radius and center.
1. (x-2)2
+(y-2)2
-36=0
2. (x+4)2
+(y-9)2
-144=0
3. x2
+y2
-2x-8y-43=0
4. x2
+y2
+4x-4y-28=0
Question:
Is there a shorter way of transforming each equation
to standard form? Share your way.
21. Solve.
The diameter of the circle is 1 unit and its center
is at (-3, 8). What is the equation of the circle?
Write the equation in standard form.
22. I. Write the equation of the following
circles given the center and the radius.
Center Radius
1 (5, 9) 49
2 (-9, 12) 64
3 (8, -25) 121
4 (-3, -27) 36
5 (0, 0) 81
6 (0, -7) 169
7 (11, 0) 144
23. II. Find the center and the radius of
the following circles.
1. (x-7)2
+(y+2)2
=9
2. x2
+(y+2)2
=25
3. (x-5)2
+y2
=36
4. x2
+y2
=49
24. III. Transform the following equations in
standard form then determine the center
and the radius.
1. x2
+y2
+10x+4y-7=0
2. x2-y2
-6x-8y-24=0
25. A radio signal can transmit messages up to a
distance of 3km. If the radio signal’s origin is located at a point
whose coordinates are (4,9), what is the equation of the circle
that defines the boundary up to which the messages can be
transmitted? Write the equation in standard form.
26. I. What defines me?
1. (x-3)2+(y-8)2=12
2. (x+6)2+(y-4)2=32
3. (x-9)2+(y+3)2=52
4. (x+1)2+(y+6)2=72
5. x2+y2=62
6. x2+(y-5)2=42
7. (x-8)2+y2=22
II. Find my Center
and Radius
1. (2, 2);6
2. (-4,9); 12
3. (2, 4); 8
4. (-2, 2); 6
III. Find Out More!
1. (x+3)2+(y-8)2=12
28. 1. Transform the equation into its standard form.
x2
+y2
+10x+4y-7=0
2. Determine the center and the radius of the following
equation.
1. x2
+ y2
=32
2. x+5)2
+ (y+9)2
=102
3. x2
+y2
+4x-4y-28=0
Let’s check your pre-test ...