5.2 Graph and Write Equations of Circles Pg180.  A circle is an infinite set of points in a plane that are equal distance away from a given fixed point.

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Presentation transcript:

5.2 Graph and Write Equations of Circles Pg180

 A circle is an infinite set of points in a plane that are equal distance away from a given fixed point called a center.  A radius is a segment that connects the center and one of the points on the circle.  Every radius is equal in length. (radii)  A diameter is a line segment that connects two points on the circle and goes through the center.

 The formula for a circle with its center at (0,0) and a radius of “r” is:  Example: What is the equation for a circle whose center is at (0,0) and has a radius of 6.  Answer ?

 Example #2: Identify the center and the radius of the following:  Answer ?  Center: (0,0)  Radius 10

 Write an equation for the following in standard form.

Graph the Equation of a Circle  Graph x 2 = 25 – y 2

Write an Equation of a Circle  The point (6, 2) lies on a circle whose center is the origin. Write the standard form of the equation of the circle.  We need to find the radius. Use the distance formula. Use the distance formula.

If a circle has a translated center then the new equation will be: Where the new center will be (h, k) and “r” will be the radius.

 Example #1. Write the standard equation for a circle whose center is at (-3,5) and has a radius of 7.  Answer: Since h=-3 and k=5 and r=7 we can substitute these into the equation  Therefore: Or:

 Example #2 Given the following equation, identify the center and the radius.  Center ?  Did you say (7,-2)  Radius ?  I bet you said (drum roll please) 9999  You guys are AWESOME !!!

 Example #3 Given the following graph answer the following questions.  1. Identify the center.  2. Name the radius.  3. Write an equation in standard form for the circle.

Answers  Center: (-2,-1)  Radius: 8  Equation:

Finding the equation of a Tangent Line to a Circle  Any line tangent to a circle will be perpendicular to a line that goes through the tangent point and the center of the circle Perpendicular lines have negative reciprocal slopes Perpendicular lines have negative reciprocal slopes (Opposite Reciprocal)(Opposite Reciprocal) So if we have the center of a circle, and the point of tangency, we can find the slope by the slope formula So if we have the center of a circle, and the point of tangency, we can find the slope by the slope formula Take the negative reciprocal and use the point slope formula for a line that goes through the tangent point Take the negative reciprocal and use the point slope formula for a line that goes through the tangent point

Find the equation of a Tangent Line to a circle  Find the equation of a tangent line to the circle x 2 + y 2 = 10 at (-1, 3)

Assignment Pg – 23 odd, 24 Pg – 17 odd, 18