4-1 Circles 3.1 Given a quadratic equation of the form ax2 + by2 + cx + dy + e = 0, use the method for completing the square to put the equation into standard form; identify whether the graph of the equation is a circle, ellipse, parabola, or hyperbola and graph the equation.

Warm Up Solve by taking the square: b2 – 6b + 9 = 25 Find the value of c that makes the function a perfect square. a2 – 12a + c Solve by completing the square c2 + 4c = 12

Essential Question What is the standard form for the equation of a circle, and what does the standard form tell you about the circle?

Explore: Deriving the Standard-Form Equation of a Circle The coordinate plane shows a circle with center C(h,k) and radius r. P(x, y) is an arbitrary point on the circle but is not directly above or below or to the left or right of C. A(x, k) is a point with the same x-coordinate as P and the same y-coordinate as C. Explain why ΔCAP is a right triangle.

Answer Since point A(x, k) is a point with the same x-coordinate as P segment PA is a vertical segment. Since point A has the same y-coordinate as C, segment CA is a horizontal segment. This means that segments PA and CA are perpendicular, which means that /_CAP is a right angle and ΔCAP is a right triangle.

Remember that point P is arbitrary, so you cannot rely upon the diagram to know whether the x-coordinate of P is greater than or less than h or whether the y-coordinate of P is greater than or less than k, so you must use absolute value for the lengths of the legs of ΔCAP. Also, remember that the length of the hypotenuse of ΔCAP is just the radius of the circle.

The length of segment AC is The length of segment AP is The length of segment CP is

Apply the Pythagorean Theorem to ΔCAP to obtain an equation of the circle.

Reflection Why isn’t absolute value used in the equation of the circle? The standard form equation of a circle with center C(h, k) and radius r is

Assignment 1

3. The circle with center C(-8, 2) and containing the point P(-1, 6)

4. Graph the circle after writing the equation in standard form. x2 + y2 – 2x – 8y + 13 = 0

5. x2 + y2 + 4x + 12y + 39 = 0

6. 2x2 + 2y2 + 20x + 12y + 50 = 0

7. A router for a wireless network on a floor of an office building has a range of 35 feet. The router is located at the point (30, 30). The lettered points in the coordinate diagram represent computers in the office. Which computers will be able to connect to the network through the router?

Write an inequality representing the problem, and draw a circle to solve the problem. 8. The epicenter of an earthquake is located at the point (2O, – 3O). The earthquake is felt up to 40 miles away. The labeled points in the coordinate diagram represent towns near the epicenter. In which towns is the earthquake felt?