The Conic Sections Chapter 10. Introduction to Conic Sections (10.1) 4 A conic section is the intersection of a plane with a double-napped cone.

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Presentation transcript:

The Conic Sections Chapter 10

Introduction to Conic Sections (10.1) 4 A conic section is the intersection of a plane with a double-napped cone.

By changing the angle and the location of the intersection, a parabola, circle, ellipse or hyperbola is produced. How are these sections created?

Circle (Section 10.2) 4 When a plane intersects a double-napped cone and is parallel to the base of a cone, a circle is formed.

Ellipse (Section 10.3) 4 When a plane intersects a double- napped cone and is neither parallel nor perpendicular to the base of the cone, an ellipse can be formed. The figure is a closed curve.

Hyperbola (Section 10.4) 4 When a plane intersects a double- napped cone and is neither parallel or perpendicular to the base of the cone, a hyperbola can be formed. The figure consists of two open curves.

Parabola (Section 10.5) 4 When a plane intersects a double- napped cone and is parallel to the side of the cone, a parabola is formed.

The Distance Formula Example: find the distance between the points (4, -2) and (8, 3) Answer is Foundational Knowledge – Chapter 10 Remember: if the answer is not a perfect square, leave as a simplified radical expression.

The Midpoint Formula ( The midpoint is the average of the two coordinates!) Example: Find the midpoint between (-2, 4) and (6, -5) … it’s (2, -1/2) The Slope Formula Example: Find the slope of the two points listed above. Foundational Knowledge

Putting it all together: the parallelogram example Example: Using a combination of these two tools, determine if the four sided shape with vertices ( -2, 3) (-3, -2) (2, -3) (3,2) is a parallelogram. Remember: to prove a parallelogram, show that either… 4 One pair of opposite sides congruent (distance) and parallel (slope) or, 4 Opposite sides are parallel (slope) or 4 Diagonals bisect each other (midpoint) or 4 Both pairs of opposite sides same length (distance)

( -2, 3) (-3, -2) (2, -3) (3,2) Compare slope and length of one pair of opposite sides Conclusion: shape is a parallelogram