Lesson-6-Equations and graph of Circles.pptx

equations of circle IN STANDARD AND GENERAL FORM

1. THE CIRCLE LESSON 6

2. CONTENTS The following will be discussed in this presentation: Chapter 6 of Analytic Geometry by Love and Rainville Definitions; General Form Equation of a Circle Reduction of the General Form to the Center-Radius Form Circle Determined by Three Conditions Tangents to a Circle

3. DEFINITION; GENERAL FORM A circle is the locus of a point that moves at a constant distance from a fixed point. The fixed point is called the Center and the constant distance is the Radius . The circle is one of the curves represented by an equation of 2 nd Degree in the form:  

4. DEFINITION; GENERAL FORM A circle is mostly conveniently represented in the general form Letting A=C and B=0. **Theorem: An equation of the second degree in which and have equal coefficient and the xy -term is missing represents a circle (exceptionally, a single point or no locus)  

5. EQUATION OF A CIRCLE Circle with Center at the Origin Circle with Center (h, k)

6. Circle with Center at the Origin Thus, the standard equation of a circle with center at the origin is   P(x, y) O y x d

7. Circle with Center at (h, k) Thus, the standard equation of a circle with center at the origin is   P(x, y) C(h, k ) h k x y O d

8. EQUATION OF A CIRCLE Example 1: Write the equation of the circle with center at (2, -3) and radius 5. Sketch the graph.

9. EQUATION OF A CIRCLE Example 2: Write the equation of the circle with center at the origin and radius 3. Sketch the graph.

10. EQUATION OF A CIRCLE Example 3: Write the equation of the circle with the points (2, 5) and (6, -1) as the endpoints of its diameter. Sketch the graph.

11. EQUATION OF A CIRCLE Example 4: Write the equation of the circle with center at the (1,1) and touching the line 3x + 4y = 10. Sketch the graph.

12. REDUCTION OF THE GENERAL FORM TO CENTER-RADIUS FORM GENERAL FORM: CENTER-RADIUS FORM:  

13. REDUCTION OF THE GENERAL FORM TO CENTER-RADIUS FORM Example 1: Find the center and radius of the circle represented by the equation 4  

14. REDUCTION OF THE GENERAL FORM TO CENTER-RADIUS FORM Example 2: Show that the circles and are tangent to each other. Sketch the graph  

15. REDUCTION OF THE GENERAL FORM TO CENTER-RADIUS FORM Example 3: Find the points of intersection of the circle and . Sketch the graph  

16. CIRCLE DETERMINED BY THREE CONDITIONS Example 1: Find the equation of the circle through the points P1:(1,1), P2:(2, -1), and P3:(2, 3).

17. TANGENTS TO A CIRCLE Line Tangent to a Circle – a line that touches the circle at exactly one point. x y O TANGENT LINE

18. TANGENTS TO A CIRCLE Example 1: Find the equation of the tangent line at (-1, 4) on the circle