Lecture Notes: Circle Equations and Related Problems

Key Concepts

• Center of a Circle: Given as ((-1, 2)).
• Point on the Circle (P): Given as ((5, 6)).
• Radius (r): Calculated using the distance formula between center ((C)) and point ((P)).

Calculating the Radius

• Distance Formula: ( \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2} )
• Substitution: ( \sqrt{(5 + 1)^2 + (6 - 2)^2} )
• Calculation: ( \sqrt{36 + 16} = \sqrt{52} )

Equation of a Circle

• General Form: ((x - h)^2 + (y - k)^2 = r^2)
• Substitution: ((x + 1)^2 + (y - 2)^2 = 52)
• Expanded Form: (x^2 + y^2 + 2x - 4y - 47 = 0)

Problem Examples

Example 1

• Center: (2, 3)
• Equation: (x^2 + y^2 + ax + by - 12 = 0)
• Radius Calculation: Using formula ( \sqrt{g^2 + f^2 - c} ) for radius.

Example 2

• Equation: (x^2 + y^2 - 4x + 6y + c = 0)
• Radius: Given as 6, find (c).

Example 3

• Endpoints of Diameter: ((4, 2), (1, 5))
• Midpoint Formula: Center is the midpoint of diameter.

Additional Problems

• Find Center and Radius: For given equations in different formats.
• Parametric Equations: Conversion to parametric form.
• Concentric Circles: Finding equation of a circle concentric with a given circle.
• Length of Tangent: From a point to a circle using distance formula.

Practical Exercises

1. Find the equation of a circle passing through given points and centers.
2. Determine the value of coefficients in circles satisfying certain conditions.
3. Solve for unknown variables in tangent and normal line equations.
4. Calculate powers and lengths related to tangents and normals in circle problems.

Summary

This lecture covered various problems and solutions related to circles in the coordinate plane. It included finding equations based on centers, radii, and points, and involved converting between different forms of circle equations. Key formulas for distance, circle equations, and tangents were applied to solve practical problems.