The Second Degree Equations

Determine whether the following equations represent a circle, parabola, ellipse or hyperbola. 1. x 2 + 4y 2 – 6x + 10y – 21 = 0 2. x 2 + y 2 – 4x + 8y – 29 = 0 3. x 2 + 9xy – 7y 2 – 2x + 6y – 100 = 0 4. x 2 – 4xy + 4y 2 – 9x + 16y – 15 = x 2 – 10y 2 – 25x + 9y + 36 = 0

The Second Degree Equations Determine whether the following equations represent a circle, parabola, ellipse or hyperbola. 6. 2x 2 + 2y 2 – 12x + 20y – 42 = x 2 + 9xy – 8y 2 – 20x + y – 20 = x 2 – 14x + 25y – 27 = x 2 – 4y x + 10y + 45 = x 2 – 4y 2 + 9x + 3y + 15 = 0

The Second Degree Equations Chapter 10 – Lesson 10.1 = The Circles Concept 1: The Center-Radius Form Concept 2: The General Form Concept 3: From General Form to Center-Radius Form Concept 4: Representation of Circles Concept 5: Circles Determined by Three Conditions Concept 6: Equations of Family of Circles Concept 7: The Radical Axis

The Second Degree Equations The Circles Definition: A circle is the set of all points on a plane that are equidistant from a fixed point. The fixed point is called the center, and the distance from the center to any point on the circle is called the radius. The center-radius form of the circle with C(h, k) and radius r

The Second Degree Equations

The Circles Problems on Circles 1.Find the equation of the circle that passes through the points (2,3), (6,1) and (4,-3). 2.Find the equation of the circle that passes through the points P 1 (1,2), P 2 (3,4) and has radius 2.

The Second Degree Equations

The Radical Axis The Properties of the Radical Axis 1. If the circles intersect at two distinct real points, the radical axis is the line containing the chord common to the two circles. 2. If two circles are tangent, then the radical axis is the common tangent to the circles at their point of tangency. 3. The radical axis of two circles is perpendicular to their line of centers. 4. All tangents drawn from a point of the radical axis are of equal length.

The Second Degree Equations