Analytic Geometry Conic Sections

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Presentation transcript:

Analytic Geometry Conic Sections Parabolas, hyperbolas, ellipses, circles

Analytic Geometry Analytic geometry, usually called coordinate geometry and earlier referred to as Cartesian geometry or analytical geometry, is the study of geometry using the principles of algebra

The Circle The plane that intersects the cone is perpendicular to the axis of symmetry of the cone.

The Ellipse The plane that intersects the cone is neither parallel nor perpendicular to the axis of symmetry of the cone and cuts through 2 “sides”

The Parabola The plane that intersects the cone is parallel to an element of the cone.

The Hyperbola The plane that intersects the cone is parallel to the axis of symmetry of the cone.

Where do you see conics in real life?

Dimensions a b 1D y x 2D y x z 3D

Ordered Pairs Review : (a,b) II (-a,b) I (a,b) III (-a,-b) IV (a,-b)

Finding the inclination of a line Θ=0 Θ Θ Θ=∏/2 Horizontal Vertical Acute Obtuse

Example: make sure you are in radian mode Find the inclination of the line 2x+3y = 6 Θ m= -a/b m= -2/3 Θ = ∏ + arctan (-2/3) Θ = ∏ +(-.588) Θ = 2.554

Distance Formula PQN is a right angled .  PQ2 = PN2 + QN2 X X’ Y’ O Y Q(x2, y2) N y2-y1 y2 Explain the derivation step by step according to animation. P(x1, y1) PQN is a right angled .  PQ2 = PN2 + QN2 y1 x1 (x2-x1)  PQ2 = (x2-x1)2+(y2-y1)2 x2

Midpoint of A(x1, y1) and B(x2,y2)