Course: Alg. 2 & Trig. Aim: Equation and Graph of Circle Do Now: What is the locus of points 4 units...

Course: Alg. 2 & Trig.Aim: Equation and Graph of Circle

Do Now: What is the locus of points 4 units from a given point?

The locus is a circle whose center is the given point and that has a radius

of 4 units.

4

444

Aim: How do we find the equation of the locus of points at a given distant from a given point?


Describe the locus of points 3 units from the from the origin.

The locus of points that are at distance d from fixed point A is a circle whose center is point A and the length of whose

radius is distance d.

The locus is a circle with a

radius of 3 units and whose center

is the origin(0,0)

3

What is the equation of this locus?

Circle Basics


Point P(x,y) is on circle O whose radius is five units. Use the distance formula to find the equation for the circle.

(0,0)

5

P(x,y)(x1 x2 )2 ( y1 y2 )2 d

(x1 0)2 ( y1 0)2 5

(x)2 (y)2 5

x2 y2 25

(4,-3)(3,-4)

(0,-5)

(-3,-4)(-4,-3)

(-5,0) (5,0)

(-4,3)(-3,4)

(0,-5)P(3,4)

(4,3)

When r = the radius of a circle, then the equation of a circle whose center is the origin (0,0) and

whose radius has a length of r is the equation

x2 + y2 = r2

32 + 42 = 52 = 25


The equation x2 + y2 = 100 represents the locus of all points at a given distance from the origin. What is the distance?

When r = the radius of a circle, then the equation of a

circle whose center is the origin (0,0) and whose radius

has a length of r is the equation

x2 + y2 = r2

r2 = 100

r = 10

radius measures 10 units

Write an equation of the locus of points that are at a distance of 5 units from the origin.

x2 + y2 = r2

x2 + y2 = 25r = 5

Model Problem


Describe fully the locus of points for the given equation.

A) x2 + y2 = 45

B) 4x2 + 4y2 = 64

The locus is a circle whose center is origin and whose radius is the square root of 45 or 3 5

The locus is a circle whose center is origin and whose radius of 4.

4x2 + y2 = 16

Is (8,6) located on the locus of a points whose distance from the origin is 10?

x2 + y2 = 100 82 + 62 = 100

64 + 36 = 100 YES


Write an equation for the locus of points that are equidistant from the circles whose equations are x2 + y2 = 16 and x2 + y2 = 64.

x2 + y2 = 16 is a circle with center at theorigin and with a radius of 4 units.

x2 + y2 = 64 is a circle with center at theorigin and with a radius of 8 units.

The locus of points would therefore be a circle whose radius is halfway between 4 and 8 units from the origin, or 6 units from the origin and whose equation is x2 + y2 = 36.

Model Problem


Point (x, y) is on a circle whose radius is r units and whose center is the point (h, k). Use the distance formula to find the equation for the circle.

r

(h,k)

(x,y)(x h)2 (y k)2 r

(x – h)2 + (y – k)2 = r2

What is the value of (h, k)? (4,3)(4, 3)

What is the equation for this circle whose radius is 5 with center at (4, 3)?

(x – 4)2 + (y – 3)2 = 52

= 5

(x – 4)2 + (y – 3)2 = 25

The standard form of an equation of a circle with

center (h, k) and radius r is

(x – h)2 + (y – k)2 = r2


Circle with Center (h,k)

The standard form of an equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2

What is the equation for the circle whose radius is 5 with center at (4, 3)?

(x – 4)2 + (y – 3)2 = 52

43

r = 5

(4,3)(h,k)

O’

x2 + y2 = 25

(0,0)O

(x – 4)2 + (y – 3)2 = 52 is a translation of x2 + y2 = 25under the rule T4,3(x,y) = (x + 4, y + 3), or

OT4 ,3 O'


Write and equation of the locus of points 6 units from the point (-3,5).

Standard equation of circle

(x – h)2 + (y – k)2 = r2

h = -3, k = 5, r = 6

(x – -3)2 + (y – 5)2 = 62

(x + 3)2 + (y – 5)2 = 36

Find the center and radius of the circle whose equation is (x + 5)2 + y2 = 25.

Does the origin lie on the locus of the points represented by the equation?

Center - (-5,0) Radius = 5

(0 + 5)2 + 02 = 25Origin -(0,0)

(5)2 = 25 YES

Model Problem


Find the coordinates of the center of the circle whose equation is

x2 + 14x + y2 + 2y = -40

Standard equation of circle(x – h)2 + (y – k)2 = r2

x2 + 14x + ( )+ y2 + 2y + ( )= -40

complete the squares

49 1 + 49 + 1

(x + 7)2 + (y + 1)2 = 10

rewrite as squares of binomials

center: (h, k) = (-7, -1)

Model Problem


Which relation is not a function?

Regents Prep

2 2

2

1) 2 4 2) 4

3) 4 4 4) 4

x y x y

x x y xy

The equation x2 + y2 – 2x + 6y + 3 = 0 is equivalent to

2 2

2 2

2 2

2 2

1) 1 3 3

2) 1 3 7

3) 1 3 7

4) 1 3 10

x y

x y

x y

x y


Determine if the point (1, 8) lies on the locus of points that are 10 units from the point (-7,2).

Model Problem


(0, 9) and (6, 1) are endpoints of a diameter of a circle. What is the equation of the circle?

Model Problem

Course: Alg. 2 & Trig. Aim: Equation and Graph of Circle Do Now: What is the locus of points 4 units...

Documents

Transcript of Course: Alg. 2 & Trig. Aim: Equation and Graph of Circle Do Now: What is the locus of points 4 units...