Circles HW #1

Complete the square and write as a squared binomial.

1.1. x2 + 6x + _____ = _____________

2.2. x2 – 10x + _____ = _____________

3.3. x2 + 24x + _____ = _____________

4.4. x2 + 2x + _____ = _____________

5.5. x2 – 5x + _____ = ______________

6.6.

This will be a critical

skill we will use later on with conic

sections!

2 3 ______ _______________x x

Solve by completing the square:

x2 + 6x – 16 = 0

x2 – 4x = 11

Write the equation in standard form for the given circle:

7616 22 yyxx 12422 yyx2 2 4 6 3 0x y x y

Lines Tangent to Circles

Section: 5.2(Green Book)

Circles Quiz: ThursdayCircles Test: Sept. 17

What we should know already:

Chord: A line segment whose endpoints are on the circle

Secant: A line that intersects the circle in two points

Radius: The distance from the center to a point on the circle

What we should know already:

Diameter: A chord that passes through the center of the circle

Tangent: A line in the plane of a circle that intersects the circle in EXACTLY one point

Helpful Theorems

If a line is tangent to a circle, then it is perpendicular to the radius drawn to the point of tangency

Helpful Theorems

In a plane, if a line is perpendicular to a radius of a circle at its endpoint on the circle, then the line is tangent to the circle.

If two lines are Perpendicular…

What do we know about their slopes?

Their slopes are opposite reciprocals to each other!!!!!!!!

Let’s use this to find some tangents….

Find an equation of the line tangent to the given circle at the given point

at (-1, 3)

What’s the slope of the radius to point (-1, 3)?

So……..

2 2 10x y

3 0

1 03

31

m

m

Slope of Tangent

The slope of the tangent must be the opposite reciprocal of -3, which is

1

3

Writing the Equation

Point Slope Form:

So to write the equation of our tangent line:

1 1( )y y m x x

13 ( 1)

31 1

33 3

1 10

3 3

y x

y x

y x

OR

You can use Slope-Intercept form:y = 1/3x + b plug in the point,(-1, 3)

to find b

Find an equation of the line tangent to the given circle at the given point

(x – 3)2 + (y + 2)2 = 130 at point (-4, 7)

Write a circular model

Cell Phones

A cellular phone tower services a 10 mile radius. You get a flat tire 4 miles east and 9 miles north of the tower. Are you in the tower’s range?

SOLUTION

STEP 1

Write an inequality for the region covered by the tower. From the diagram, this region is all points that satisfy the following inequality:

x2 + y2 < 102

In the diagram above, the origin represents the tower and the positive y-axis represents north.

Write a circular model continued….

STEP 2

Substitute the coordinates (4, 9) into the inequality from Step 1.

x2 + y2 < 102 Inequality from Step 1

42 + 92 < 102? Substitute for x and y.

The inequality is true.97 < 100

ANSWER

So, you are in the tower’s range.

HOMEWORK

Circle Homework #2 WS