8.3 GRAPH AND WRITE EQUATIONS OF CIRCLES

Transcription

1 8.3 GRAPH AND WRITE EQUATIONS OF CIRCLES What is the standard form equation for a circle? Why do you use the distance formula when writing the equation of a circle? What general equation of a circle is used when the center of the circle is translated? How do you put an equation into standard form?

2 Standard Equation of a Circle OP = r P(x,y) r O An equation for the circle with its center at (0,0) and a radius of r is

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

4 Graph the equation. Identify the radius of the circle. 1. x + y = 9 SOLUTION Equation is in the standard form x + y = 9 STEP 1 Identify the Center and radius form the equation, the graph is a circle centered at the origin with radius r = 9 = 3. STEP Draw the circle. First plot several convenient points that are 3 units from the origin, such as (0, 3), (3, 0), (0, 3), and ( 3, 0). Then draw the circle that passes through the points.

5 SOLUTION. y = x + 49 STEP 1 Rewrite the equation y = x + 49 in standard form as x + y = 49. STEP Identify the center and radius. From the equation, the graph is a circle centered at the origin with radius r = 49 = 7. STEP 3 Draw the circle. First plot several convenient points that are 7 units from the origin, such as (0, 7), (7, 0), (0, 7), and ( 7, 0). Then draw the circle that passes through the points.

6 Write an equation of a circle. The point (, 5) lies on a circle whose center is the origin. Write the standard form of the equation of the circle. SOLUTION Because the point (, 5) lies on the circle, the circle s radius r must be the distance between the center (0, 0) and (, 5). Use the distance formula. r = ( 0) + ( 5 0) = The radius is Use the standard form with r = equation of the circle. x + y = r = ( 9 ) x + y x + y = = 9 to write an Standard form Substitute 9 for r Simplify 9

7 4. Write the standard form of the equation of the circle that passes through (5, 1) and whose center is the origin. SOLUTION Because the point (5, 1) lies on the circle, the circle s radius r must be the distance between the center (0, 0) and (5, 1). Use the distance formula. r = (5 0) + ( 1 0) = 6 Use the standard form with r = equation of the circle. x + y = r 6 to write an Standard form x + y x + y = 6 The radius is Simplify 6

8 SOLUTION A line tangent to a circle is perpendicular to the radius at the point of tangency. Because the radius to the point ( 3, ) has slope 0 m = 3 0 = 3 the slope of the tangent line at (3, ) is the negative reciprocal of or 3 An equation of 3 the tangent line is as follows: y = 3 3 (x ( 3)) y = x + y = 3 x Point-slope form Distributive property Solve for y. The correct answer is C.

9 5. Write an equation of the line tangent to the circle x + y = 37 at (6, 1). SOLUTION A line tangent to a circle is perpendicular to the radius at the point of tangency. Because the radius to the point (6, 1) has slope m = 1 0 = the slope of the tangent line at (6, 1) is the negative reciprocal of 1 or 6 An equation of 6 the tangent line is as follows: Point-slope form y 1 = 6(x 6) y 1 = 6x + 36 y = 6x + 37 Distributive property Solve for y.

10 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 In the diagram above, the origin represents the tower and the positive y-axis represents north. 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 + y < 10 STEP x 4 Substitute the coordinates (4, 9) into the inequality from Step 1. + y < 10? + 9 < < 100 Inequality from Step 1 Substitute for x and y. So, you are in the The inequality is true. tower s range.

11 Standard Equation of a Circle The standard equation for a translated circle is (x h) + (y k) = r center: (h, k) radius: r

12 Example Write the standard equation of the circle graphed below. Where is the center? (h,k) = (-,3) What is the radius? r=

13 Practice Write the standard equation of a circle with the following center and radius. (x h) + (y k) = r 1) C(0,0) radius: 9 ) C(,3) radius: 5 3) C(-5,) radius: 4 center (h,k)

14 Practice Graph each equation. Label the center and radius. (x h) + (y k) = r center (h,k) 1) x + y = 5 ) (x ) + y = 4 3) (x + 4) + (y 3) = 49

15 Example Write the standard equation for the circle given by x + y 1x y - 8 = 0. State the coordinates of its center and give its radius. COMPLETE THE SQUARE! Center: (6,1) Radius:

16 Example Write the standard equation for the circle given by x + y + 6x 4y - 3 = 0. State the coordinates of its center and give its radius. Then sketch the graph. 8 6 Center: (-3,) 4 Radius:

17 Practice Write the standard equation for the circle given by x + y - x + y - 7 = 0. State the coordinates of its center and give its radius. Then sketch the graph Complete the square Factor Center is at (1, 1), r =3

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19 Hw 8.3 p. 505, 4-18 even, - 8 even, 54, 56 p. 531, 4, 8, 13, 14