Overview of topics: Definition of a circle to orthogonality.
Preparation for the next lecture on parabolas.
Definition of a Circle
Locus Definition: A circle is defined as the locus of a point which moves in a plane such that its distance from a fixed point (the center) always remains constant.
Key Terms:
Center: Fixed point in the plane.
Radius: Fixed distance from the center.
Equation of a Circle
Circle centered at (0,0):
Equation: ( x^2 + y^2 = r^2 )
Circle centered at (h,k):
Equation: ( (x - h)^2 + (y - k)^2 = r^2 )
General Equation of Circle:
Form: ( x^2 + y^2 + 2gx + 2fy + c = 0 )
Center: ( (-g, -f) )
Radius: ( \sqrt{g^2 + f^2 - c} )
Nature of Circles
Real Circle: ( g^2 + f^2 - c > 0 )
Point Circle: ( g^2 + f^2 - c = 0 )
Imaginary Circle: ( g^2 + f^2 - c < 0 )
Intercepts of a Circle
X-Intercept: ( \sqrt{g^2 - c} )
Y-Intercept: ( \sqrt{f^2 - c} )
Conditions:
If ( g^2 - c > 0 ): Two real intercepts.
If ( g^2 - c = 0 ): Circle touches the axis.
If ( g^2 - c < 0 ): No intercepts.
Tangents and Normals
Tangent Definition: Line that touches the circle at one point.
Normal Definition: Line perpendicular to the tangent at the point of tangency.
Key Information:
Normal passes through the center of the circle.
Conditions to find relationships between a line and a circle:
Find the length of the perpendicular from the center to the line.
Common Tangents of Circles
Types:
Direct Common Tangent (DCT)
Transverse Common Tangent (TCT)
Determinations:
If centers are on the same side of the tangent, it's DCT.
If centers are on opposite sides, it's TCT.
Length of Common Tangents:
DCT: ( \sqrt{d^2 - (r_1 - r_2)^2} )
TCT: ( \sqrt{d^2 - (r_1 + r_2)^2} )
Orthogonality of Circles
Condition for Orthogonality: Two circles are orthogonal if ( g_1 g_2 + f_1 f_2 = c_1 + c_2 ).
Radical Axis and Radical Center
Radical Axis: Locus of points that have equal power with respect to two circles.
Radical Center: Point where the radical axes of three circles intersect.
Conclusion
Review of radical axis properties and applications in finding orthogonal circles.
Encouragement to complete notes and prepare for the next session.