Name Period Date: Topic: 9-2 Circles Essential Question: If the coefficients of the x 2 and y 2 terms in the equation for a circle were different, how would that change the shape of the graph of the equation? Standard: G-GPE.1 Objective: Derive the equation of a circle of given center and radius using the Pythagorean Theorem; complete the square to find the center and radius of a circle given by an equation. To learn the relationship between the center and radius of a circle and the equation of the circle. Analytic geometry uses algebra to investigate geometric figures. In Lessons 9-2 through 9-6 you will study plane curves having second-degree equations. These curves are called conic sections, or simply conics, because they can be obtained by slicing a double cone with a plane (see figure). Summary
The diagram shows a circle having center and radius 6. To find the equation of the circle, use the distance formula. A point is on the circle if and only if the distance between P and the center equals 6. = + + = + + = Every circle has an equation of this form. Equation of a Circle: Equation of a Circle The circle with center (h, k) and radius r has the equation x h 2 + y k 2 = r 2 Example 1: Find an equation of the circle with center and radius 3. Solution Substitute =, =, and = in this equation: + = [ ] + = + + = Answer If the center of a circle is at the origin, then is and the equation of the circle is, + =, or + =. 2
Exercise 1: Find an equation of the circle with center and radius 3. Find an equation of the circle with center and radius. 3
Translation: The graph of the equation + = is a circle with center at the origin and radius 3. If you slide every point of this circle to the right 2 units and up 5 units, the equation of the translated circle is + =. Sliding a graph to a new position in the coordinate plane without changing its shape is called a translation. In general, replacing A by x - h and y by y ~ k in an equation slides the corresponding graph h units horizontally and k units vertically. Slide: right if h is positive, left if h is negative, up if k is positive, down if k is negative. Example 2: Solution 1 Graph + + = The graph of the equation + = is a circle with center at the origin and radius 2. Slide this circle 2 units to the right and then 6 units down to get the graph of + =, or + + =. Solution 2 Rewrite the given equation in the form + = + = The graph is a circle. The center is and the radius =. 4
Exercise 2: Graph + = Graph + + = 5
Example 3: If the graph of the given equation is a circle, find its center and radius. If the equation has no graph, say so. + + + = If the graph is a circle, the equation can be written in the form + = Complete the square twice, once using the terms in x, and once using the terms in y. ( + + ) + ( + ) = ( + + ) + ( + ) = + + + + = the center is and the radius is. Answer If the graph of the given equation is a circle, find its center and radius. If the equation has no graph, say so. + + + = ( + ) + ( + + ) = ( + ) + ( + + ) = + + + + = Since the square of any real number is positive and the sum of two positive numbers is positive, no ordered pair satisfies the equation. This equation has no graph. Answer 6
Exercise 3: If the graph of the given equation is a circle, find its center and radius. If the equation has no graph, say so. + + = If the graph of the given equation is a circle, find its center and radius. If the equation has no graph, say so. + + + + = 7
Recall this fact from geometry: Let L be the line tangent to a given circle at a point P. Then the line perpendicular to L at P passes through the center of the circle. Example 4: Find an equation of a circle of radius 3 that has its center in the first quadrant and is tangent to the y-axis at. Solution First make a sketch using the given information. From the geometric fact stated above, the center of the circle lies on the line = and must be 3 units from, which is on the circle. Thus the center is. The equation is + = Answer Exercise 4: Find an equation of a circle that has center the point. and contains 8
Exercise 4 continued: Find an equation of a circle that has a diameter with endpoints and. (A sketch may be helpful.) 9
Exercise 4 continued: Find an equation of a circle that has its center in the quadrant four and is tangent to the lines =, =, and =. (A sketch may be helpful.) Class work: p 409 Oral Exercises: 1-15 Homework: p 410 Written Exercises: 2-14 even p 410 Written Exercises: 16-42 even 10