LESSON 1 INTRODUCTION TO CONIC SECTIONS AND CIRCLES

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LESSON 1: INTRODUCTION TO CONIC SECTIONS AND CIRCLES

LESSON 1: INTRODUCTION TO CONIC SECTIONS AND CIRCLES

OBJECTIVE/S v. Illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and

OBJECTIVE/S v. Illustrate the different types of conic sections: parabola, ellipse, circle, hyperbola, and degenerative cases. v. Define a circle v. Determine the equation of a circle in standard form; and v. Sketch a circle in a rectangular coordinate system.

RECALL

RECALL

o What are other terms for rectangular coordinate system? o How are points labelled

o What are other terms for rectangular coordinate system? o How are points labelled in the coordinate system? o When is a point solution of the given equation?

FUNDAMENTAL THEOREM OF ANALYTIC GEOMETRY There exists a one-to-one correspondence between the geometric plane

FUNDAMENTAL THEOREM OF ANALYTIC GEOMETRY There exists a one-to-one correspondence between the geometric plane and the set of ordered pairs of real numbers. (i. e. R 2 )

SLOPE •

SLOPE •

DISTANCE FORMULA •

DISTANCE FORMULA •

MIDPOINT FORMULA •

MIDPOINT FORMULA •

ANALYZE AND EXPLORE

ANALYZE AND EXPLORE

CONIC SECTION/CONIC Double Right Circular Cone– the result of rotating a line about the

CONIC SECTION/CONIC Double Right Circular Cone– the result of rotating a line about the vertex in such a way that its angle with the axis is fixed. Axis of the cone– the line of symmetry of the cone which passes through the vertex.

CONIC SECTION/CONIC --- is the graph of a second-degree equation in the coordinates x

CONIC SECTION/CONIC --- is the graph of a second-degree equation in the coordinates x and y. --- is the curve produced by cutting or slicing the surface of cone/s with a plane.

CONIC SECTION/CONIC q. Ellipse q. Parabola q. Hyperbola v Circle – special type of

CONIC SECTION/CONIC q. Ellipse q. Parabola q. Hyperbola v Circle – special type of ellipse

DEGENERATE CONICS When a plane intersects the double right circular cone at its vertex,

DEGENERATE CONICS When a plane intersects the double right circular cone at its vertex, q. Ellipse becomes a point q. Parabola becomes a line q. Hyperbola becomes 2 intersecting lines

CIRCLE Is a locus of all points in the plane having the same fixed

CIRCLE Is a locus of all points in the plane having the same fixed positive distance, called the radius, from a fixed point, called the center.

CIRCLE

CIRCLE

• THEOREM: EQUATION OF A CIRCLE IN STANDARD FORM

• THEOREM: EQUATION OF A CIRCLE IN STANDARD FORM

THEOREM: EQUATION OF A CIRCLE IN STANDARD FORM The general form of the equation

THEOREM: EQUATION OF A CIRCLE IN STANDARD FORM The general form of the equation of a circle: x 2 + y 2 + Dx + Ey + F = 0

• THEOREM: EQUATION OF A CIRCLE IN STANDARD FORM

• THEOREM: EQUATION OF A CIRCLE IN STANDARD FORM

EXAMPLES: 1. Find an equation of the circle with the following conditions. a. Center

EXAMPLES: 1. Find an equation of the circle with the following conditions. a. Center is at C(2, -5) and radius 3. b. Has a diameter whose endpoints are the points A(3, 4) and B(-3, 12). c. Center is at C(-10, 0) and passes through A(-6, 3). d. Center is at C(-4, 6) and is tangent to the x-axis.

EXAMPLES: 2. Sketch the graph of the following equations: a. x 2 + y

EXAMPLES: 2. Sketch the graph of the following equations: a. x 2 + y 2 + 10 x -2 y = 55 b. x 2 + y 2 + 4 x + 6 y -23 =0 c. x 2 + y 2 + 6 x – 10 y + 40 = 0 d. x 2 + y 2 + 6 x -10 y + 34 = 0

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