11.0 Analytic Geometry & Circles

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Presentation transcript:

11.0 Analytic Geometry & Circles Objectives: Define and identify a conic section. Define analytic geometry. Define a circle. Write the equation of a circle. Identify important characteristics of circles. Graph circles.

Conic Sections Conic Sections – The geometric figures formed by slicing a right circular double-napped cone at different angles with a plane. Analytic Geometry – The study of geometric properties of objects using the coordinate system.

Circles A circle is a special type of ellipse where all the points in it are equidistant from a given point called the center. The distance from the center to any point on the circle is called the radius. The Standard Form for the equation of a circle:

Circles By centering the circle at the origin we obtain:

Example #1 Graph and describe the circle whose equation is:

Example #1 Graph the circle with center (1, -3) and radius 4, and find its equation.

Example #2 Graph the circle with center (4, -2) that passes through (3,5), and find its equation.

Example #3 Write the equation of the circle into standard form and graph.

Example #3 Write the equation of the circle into standard form and graph.

Example #3 Write the equation of the circle into standard form and graph.

Example #3 Write the equation of the circle into standard form and graph.

Example #4 Find the equation of a circle tangent to the y-axis (meaning it touches in a single point) that is centered at (5, -1). Since it is tangent to the y-axis, the radius is 5.

Example #5 Find the equation of a circle whose endpoints of the diameter are (4, -6) & (-1, -2). Since the endpoints of the diameter are given, the center can be found by using the midpoint formula. The radius is then half the distance between the two endpoints.