Conic Sections - Parabola

Questions and Answers

What defines a parabola in geometric terms?

The set of points equidistant from a point and a straight line. (correct)

The locus of points equidistant from two parallel lines.

The set of points equidistant from two foci.

The intersection of a cone with a plane intersecting both nappes.

If the equation of a parabola is given as $y^2 = -12x$, what can be determined about its orientation?

It opens upward.

It opens to the right.

It opens to the left. (correct)

It opens downward.

In the equation $x^2 = -10y$, what does the negative value of $p$ indicate about the parabola?

It opens upward.

It opens leftward.

It opens to the right.

It opens downward. (correct)

In what form is the equation of a parabola expressed when the vertex is at the point (h, k) and the focus at (h + p, k)?

$(y - k)^{2} = 4p(x - h)$ Signup and view all the answers

Which of the following describes the axis of symmetry of a parabola?

A vertical line that is perpendicular to the directrix passing through the focus. Signup and view all the answers

What is the vertex of the parabola given by the equation $y^2 = 6x$?

(0, 0) Signup and view all the answers

If a parabola opens upward, what can be said about the value of p in the equation $x^2 = 4py$?

p must be greater than zero. Signup and view all the answers

What is the directrix of the parabola defined by the equation $y^2 = 4px$ for p = 2?

x = -2 Signup and view all the answers

Which of the following parabolas has a vertex located at the origin and opens to the left?

$y^2 = -4x$ Signup and view all the answers

What property defines the distance from the vertex to the focus and the vertex to the directrix in a parabola?

They are equal distances. Signup and view all the answers

Study Notes

Definition and Characteristics

A parabola is formed by the intersection of a plane with one nappe of a cone.
It is defined as the set of points equidistant from a point (focus) and a line (directrix).
The axis of symmetry is a line perpendicular to the directrix that passes through the focus.

Key Components

Focus: A specific point where all reflective paths converge.
Directrix: A fixed line used in the definition of a parabola.
Vertex: The point where the axis of symmetry intersects the parabola.

Equations of Parabolas

Horizontal Parabola (Vertex at Origin):
- Equation: ( y^2 = 4px )
- Opens to the right if ( p > 0 ); to the left if ( p < 0 ).
Vertical Parabola (Vertex at Origin):
- Equation: ( x^2 = 4py )
- Opens upward if ( p > 0 ); downward if ( p < 0 ).

Vertex Form of Parabola

Equation: ( (y - k)^2 = 4p(x - h) )
Focus located at ( (h + p, k) ).
Opens to the right if ( p > 0 ); to the left if ( p < 0 ).

Example Problems

Analyze and sketch the parabolas given in the equations:
- ( y^2 = -12x )
- ( 2x^2 - 14y = 0 )
- ( y^2 = 6x )
- ( x^2 = -10y )
- ( 2y^2 = 9x )
- ( y^2 + 7x = 0 )
- ( 3x^2 - 4y = 0 )

Determining Key Features

For each parabola, identify:
- Vertex
- Focus
- Directrix
- Axis of symmetry
Include these components in the sketch of the graph.

Description

Explore the fascinating world of parabolas in this quiz on conic sections. Understand their unique characteristics, definitions, and geometric properties, including focus and directrix. Test your knowledge and learn more about the shape formed from the intersection of a plane with a cone.