9.3 Graph and Write Equations of Circles

 

Circle

The set of all points (x, y) that are equidistant from a fixed point

 

Center

The fixed point that is equidistant from all the points on a circle

 

Radius

The distance r between the center and any point (x, y) on a circle

 

STANDARD EQUATION OF A CIRCLE WITH CENTER AT THE ORIGIN

The standard form of the equation of a circle with center at (0, 0) and radius r is as follows:

x2 + y2 = _r2_

 

Example 1

 

Graph y2 = -x2 + 16. Identify the radius of the circle.

 

Solution

1.      Rewrite the equation y2 = -x2 + 16 in standard form as _x2 + y2 = 16_.

2.      Identify the center and radius. From the equation, the graph is a circle centered at the origin with radius r =          = 4.

3.      Draw the circle. First plot several convenient points that are 4 units from the origin, such as (0, _4_), (4, _0_), (0, _-4_ ), and (-4, _0_). Then draw the circle that passes through the points.

 

 

 


Example 2

Write an equation of a circle

 

The point (-3, 4) lies on a circle whose center is the origin. Write the standard form of the equation of the circle.

 

The circle's radius r must be the distance between the center and (-3, 4). Use the distance formula.

 


r =

=                  =              =_5_

Use the standard form with r = _5_ to write an equation of the circle.

 

x2 + y2 = r2

Standard form

x2 + y2 = _5_2

Substitute for r.

x2 + y2 = _25_

Simplify.

 

Example 3

Find a tangent line

 

Write an equation of the line tangent to the circle x2 + y2 = 17 at (4, -1).

 

A line tangent to a circle and the radius to the point of tangency are perpendicular. The radius with endpoint (4, -1) has slope m =                                                   , so the slope of the tangent line at (4, -1) is the negative reciprocal of                                                           , or _4_. An equation of the tangent line is as follows:

y + _1_ = _4_(x - _4_)

Point-slope form

y = _4x - 17_

Solve for y.

 

 

 

 

 

 

 

 

Example 4

 

Lighthouse The beam from Oak Island Lighthouse in North Carolina can be seen for up to 24 miles. You are 18 miles east and 9 miles south of the lighthouse. Can you see the lighthouse beam?

 

Solution

1.  Write an inequality for the region lit by the beam. This region is all the points that satisfy the following inequality: x2 + y2 < _24_2

2.  Substitute the coordinates (18, 9) into the inequality.

 

x2 + y2 < _24_ 2

Inequality

_182 + 92_ < _24_ 2

Substitute for x and y.

_405 < 576_

The inequality is _true_ .

You _can_ see the lighthouse beam.