1. Do Now: What is the locus of points 4 units from a given point?
Aim: How do we find the equation of the locus of points at a given distant from a given point?
Do Now: What is the locus of points 4 units from a given point?
LOCUS 4
Fundamental Locus 5 4 4 4
The locus is a circle whose center is the given point and that has a radius of 4 units.

2. Describe the locus of points 3 units from the from the origin.
Circle Basics
Describe the locus of points 3 units from the from the origin.
The locus is a circle with a radius of 3 units and whose center is the origin 3
(0,0)
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.
Fundamental Loci -5
What is the equation of this locus?

3. Point P(x,y) is on circle O whose radius is five units
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,-5)
P(x,y)
P(3,4)
(-3,4) 5
(4,3)
(-4,3)
(-5,0)
(5,0)
(0,0)
(-4,-3)
(-3,-4)
(4,-3)
= 52 = 25
(3,-4)
(0,-5)
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

4. Model Problem
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
r = 5
x2 + y2 = 25

5. YES 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
x2 + y2 = 16 4
The locus is a circle whose center is origin and whose radius of 4.
Is (8,6) located on the locus of a points whose distance from the origin is 10?
x2 + y2 = 100
= 100
YES
= 100

6. Model Problem
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 the
origin and with a radius of 4 units.
x2 + y2 = 64 is a circle with center at the
origin 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.

7. 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.
(x,y) r
= 5
(x – h)2 + (y – k)2 = r2
(h,k)
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)?
The standard form of an equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2
(x – 4)2 + (y – 3)2 = 52
(x – 4)2 + (y – 3)2 = 25

8. Circle with Center (h,k)
What is the equation for the circle whose radius is 5 with center at (4, 3)?
(x – 4)2 + (y – 3)2 = 52 r
= 5
(4,3)
(h,k) O’
The standard form of an equation of a circle with center (h, k) and radius r is (x – h)2 + (y – k)2 = r2 4 3
x2 + y2 = 25
(0,0) O
(x – 4)2 + (y – 3)2 = 52 is a translation of x2 + y2 = 25
under the rule T4,3(x,y) = (x + 4, y + 3), or

9. Standard equation of circle
Model Problem
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
Origin -(0,0)
YES
(0 + 5) = 25
(5)2 = 25

10. Standard equation of circle
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
complete the squares
x2 + 14x + ( )+ y2 + 2y + ( )= -40 49 1
rewrite as squares of binomials
(x + 7)2 + (y + 1)2 = 10
center: (h, k) = (-7, -1)

11. Which relation is not a function?
Regents Prep
Which relation is not a function?
The equation x2 + y2 – 2x + 6y + 3 = 0 is equivalent to

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

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