The Second Degree EquationsGeneral Form of The Second Degree Equations

Identification of Conics given the equationsRule #1: If xy term exists, meaning B 1. If 2. If it is a parabola.3. If , it is an ellipse.

The Second Degree EquationsIdentification of Conics given the equationsRule #2: If xy term does not exist, meaning B 1. If it is a circle.2. If it is a parabola.3. If , it is an ellipse.4. If , it is hyperbola.

The Second Degree EquationsDetermine whether the following equations represent a circle, parabola, ellipse or hyperbola.

1. x2 + 4y2 – 6x + 10y – 21 = 02. x2 + y2 – 4x + 8y – 29 = 03. x2 + 9xy – 7y2 – 2x + 6y – 100 = 04. x2 – 4xy + 4y2 – 9x + 16y – 15 = 05. 4x2 – 10y2 – 25x + 9y + 36 = 0

The Second Degree EquationsDetermine whether the following equations represent a circle, parabola, ellipse or hyperbola.

6. 2x2 + 2y2 – 12x + 20y – 42 = 07. 5x2 + 9xy – 8y2 – 20x + y – 20 = 08. 10x2 – 14x + 25y – 27 = 09. 9x2 – 4y2 + 10x + 10y + 45 = 010. 4x2 – 4y2 + 9x + 3y + 15 = 0

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

The Second Degree EquationsThe 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

222 )( kyhxr

The Second Degree EquationsThe CirclesIf the center of the circle is at the origin, , and the radius is r, its equation becomes,

Write the equation of the circles given1. the center of the circle is at (3,-2) and the radius is 42. the center of the circle is at (10,0) and the radius is 33. P1(0,0) and P2(-8,6) are the ends of its diameter

The Second Degree EquationsThe CirclesThe General Form for the equation of a circle is

1. Write the equation of the circle in its general form given .

2. Write the equation of the circle in its general form with .

Write the equation in the Center-Radius Form1. 2.

The Second Degree EquationsThe CirclesNot all equations of the form

1. 2. 3.

represent a circle. If the radius is the equation represents a point circle. , the equation represents a real circle.

, the equation represents imaginary circle.

The Second Degree EquationsThe CirclesDetermine the representation of the circle defined by the following equations.

1. 2. 3.

The Second Degree EquationsThe CirclesProblems on Circles1. 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 P1(1,2), P2(3,4) and has radius 2.

The Second Degree EquationsThe Family of CirclesGiven:

by taking k as a parameter, the equation of the family of circles is

= 0

The Second Degree EquationsThe Family of Circles1. Write the equation of the family of circles represented

by the equations and and find the member of the family which passes through the point P(6, 1).

2. Graph the circles C1 and C2 whose equations are

C1:

C2:

also graph the member C3 of the family of circles for k = 1.

The Second Degree EquationsThe Radical AxisThe equation of the family of circles with k = -1, is = 0

Simplifying the equation, we have (1)

(1) is known as the radical axis of two non-concentric circles. Note: The radical axis of two non-concentric circles is a straight line obtained by taking the difference between the two circles.

The Second Degree EquationsThe Radical AxisThe Properties of the Radical Axis1. 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 EquationsThe Radical AxisGiven the intersecting circles whose equations are and

1. Find the equation of the radical axis2. Find the points of intersection of the circles3. Show that the radical axis is perpendicular to the line of centers