Circles 5.3 (M3)

EXAMPLE 1 Graph an equation of a circle Graph y 2 = – x Identify the radius of the circle. SOLUTION STEP 1 Rewrite the equation y 2 = – x in standard form as x 2 + y 2 = 36. STEP 2 Identify the center and radius. From the equation, the graph is a circle centered at the origin with radius r = 36 = 6.

EXAMPLE 1 Graph an equation of a circle STEP 3 Draw the circle. First plot several convenient points that are 6 units from the origin, such as (0, 6), (6, 0), (0, –6), and (–6, 0). Then draw the circle that passes through the points.

EXAMPLE 2 Write an equation of a circle The point (2, –5) lies on a circle whose center is the origin. Write the standard form of the equation of the circle. SOLUTION Because the point (2, –5) lies on the circle, the circle’s radius r must be the distance between the center (0, 0) and (2, –5). Use the distance formula. r = (2 – 0) 2 + (–5 – 0) 2 = 29 = The radius is 29

EXAMPLE 2 Write an equation of a circle Use the standard form with r to write an equation of the circle. = 29 x 2 + y 2 = r 2 Standard form x 2 + y 2 = ( 29 ) 2 Substitute for r 29 x 2 + y 2 = 29 Simplify

EXAMPLE 3 Standardized Test Practice SOLUTION A line tangent to a circle is perpendicular to the radius at the point of tangency. Because the radius to the point (1–3, 2) has slope = 2 – 0 – 3 – 0 = 2 3 – m

EXAMPLE 3 Standardized Test Practice 2 3 – the slope of the tangent line at (–3, 2) is the negative reciprocal of or An equation of 3 2 the tangent line is as follows: y – 2 = (x – (– 3)) 3 2 Point-slope form 3 2 y – 2 = x Distributive property y = x + Solve for y. ANSWER The correct answer is C.

GUIDED PRACTICE for Examples 1, 2, and 3 Graph the equation. Identify the radius of the circle. 1. x 2 + y 2 = 9 SOLUTION 3

GUIDED PRACTICE for Examples 1, 2, and 3 2. y 2 = –x SOLUTION 7

GUIDED PRACTICE for Examples 1, 2, and 3 3. x 2 – 18 = –y 2 SOLUTION 2

GUIDED PRACTICE for Examples 1, 2, and 3 4. Write the standard form of the equation of the circle that passes through (5, –1) and whose center is the origin. SOLUTION x 2 + y 2 = Write an equation of the line tangent to the circle x 2 + y 2 = 37 at (6, 1). y = –6x + 37 SOLUTION

EXAMPLE 4 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: x 2 + y 2 < 10 2 In the diagram above, the origin represents the tower and the positive y -axis represents north.

EXAMPLE 4 Write a circular model STEP 2 Substitute the coordinates (4, 9) into the inequality from Step 1. x 2 + y 2 < 10 2 Inequality from Step < 10 2 ? Substitute for x and y. The inequality is true. 97 < 100  ANSWER So, you are in the tower’s range.

Write the standard form of the circle Complete the square to create the standard form Complete the square to create the standard form

Write the standard form of the circle