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Korolek [52]
2 years ago
13

THIS IS NOT A TEST OR ASSESSMENT!! NO LINKS OR ANSWERING QUESTIONS YOU DON'T KNOW!!! PLEASE EXPLAIN!! Chapter 13

Mathematics
2 answers:
Alex2 years ago
4 0

Answer:

The vertex of the parabola is at equal distance between focus and the directrix. If F is the focus of the parabola, V is the vertex and D is the intersection point of the directrix and the axis of symmetry, then V is the midpoint of the line segment ¯FD

Step-by-step explanation:

answer 3rd

Andreas93 [3]2 years ago
4 0

9514 1404 393

Explanation:

<h3>1.</h3>

A cone is a 3-dimensional object created by revolving a line about an axis that intersects that line. This figure is a "double-napped" cone. The point where the revolved line and the axis meet is the a.pex, or vertex, of the cone. Typically, we're concerned with a finite portion of the cone, from the vertex to a base that is a circle in a plane perpendicular to the axis.

A "conic" is a 2-dimensional figure that results from the intersection of a plane and a cone. There are four general categories, named according to the angle the plane makes with the axis and/or the side of the cone. These are illustrated in the attachment.

  1. a circle - the plane of intersection is perpendicular to the axis
  2. an ellipse - the plane of intersection is at an angle between 90° and the angle of the side relative to the axis. Both an ellipse and a circle are closed figures.
  3. a parabola - the plane of intersection is at the same angle as the side of the cone. A parabola is a one-sided open figure.
  4. a hyperbola - The plane of intersection is at an angle between that of the side of the cone and the axis of the cone. The plane will intersect both parts of a double-napped cone producing a double-sided open figure.

Producing these at home can be an interesting project. A circle can be made using a compass.

An ellipse can be drawn using a pair of pins and a loop of string. The pins would be placed at the foci of the ellipse, and the string would constrain the drawing instrument (pen or pencil) to have a constant total distance to the two foci.

A parabola can be drawn on graph paper using coordinates derived from an equation for it. It can also be drawn using a compass and a set square by plotting points that are equidistant from the <em>focus</em> and a line that is called the <em>directrix</em>. If you have a physical cone-shaped object, you can cut it at an angle that will produce a parabola.

A hyperbola can be drawn on graph paper from an equation. It can also be drawn using a compass by plotting points that have a constant difference in their distance to the two foci, or by plotting points whose ratio of distance to focus and directrix is a constant. A physical cone-shaped object can be cut to produce a hyperbola.

__

<h3>2.</h3>

The <em>general form</em> equation for a conic is ...

  Ax² +Bxy +Cy² +Dx +Ey +F = 0

Usually, we're concerned with conics that have axes parallel to the coordinate axes, so B=0. The equation of an ellipse has A and C with the same sign. The equation of a hyperbola has A and C with opposite signs,

In <em>standard form</em>, the equations for figures centered at the origin are ...

  • ellipse: x²/a +y²/b = 1
  • hyperbola: x²/a -y²/b = 1 (opens horizontally)
  • hyperbola: y²/a -x²/b = 1 (opens vertically)

__

<h3>3.</h3>

The vertex of a conic is an extreme point on the (major) axis of the conic. The focus is a point used in the definition of the conic. The focus is "inside" the curve, on the axis of symmetry. The directrix is a line used in the definition of the conic. The directrix is "outside" the curve, perpendicular to the axis. The second attachment shows these for a parabola.

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So you can see that our vertex is (-4,-3)  
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