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6-2 Conic Sections: Circles

6-2 Conic Sections: Circles. Geometric definition: A circle is formed by cutting a circular cone with a plane perpendicular to the symmetry axis of the cone. This intersection is a closed curve, and the intersection is parallel to the plane generating the circle of the cone.

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6-2 Conic Sections: Circles

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  1. 6-2 Conic Sections: Circles Geometric definition: A circle is formed by cutting a circular cone with a plane perpendicular to the symmetry axis of the cone. This intersection is a closed curve, and the intersection is parallel to the plane generating the circle of the cone. Algebraic definition: A circle is the set of all points that are equally distant from a fixed point (the center).

  2. Center-Radius Form Example 1: Find the center and radius: (x,y) r (h,k) x y

  3. Example 2: Find the center and radius: Example 3: Find the center and radius: Example 4: Find the center and radius:

  4. Example 5: Write the equation of a circle centered at (2, 7) and having a radius of 5. Example 6: Describe Example 7: Describe

  5. Example 8: Rewrite in center-radius form by completing the square in x and y: Example 9: Rewrite in center-radius form by completing the square in x and y:

  6. Example 10: Find an equation of the line tangent to circleat point P(3, 5). Step 1: Write in center-radius form: Step 2: Check that P(3, 5) lies on the rim of the circle: Step 3: Identify the center and radius: Step 4: Find slope of radius from center (2, 0) to P(3, 5): Step 5: Write equation in point-slope form:

  7. Example 11: Find the intersection points between the circle and the line : Step 1: Solve the linear equation for one variable: Step 2: Substitute into variable of circle equation: Step 3: Substitute single-variable solutions into linear equation to solve for corresponding values:

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