LECTURE 3 PARABOLA

Definition of a Parabola
Standard Equation of a Parabola
The standard form of the equation of a parabola depends on its orientation
For a parabola that opens upwards or downwards
The focus is at (a, 0) and the directrix is the line x = -a.
For a parabola that opens left or right
The focus is at (0, a) and the directrix is the line y = -a.
The vertex is at the origin (0, 0) in these cases.
Focus, Directrix, and Axis of Symmetry
Focus: - The fixed point that helps define the parabola. It determines the direction and the distance of the curve from the vertex.
Directrix: - The fixed line used along with the focus to define the parabola. Every point on the parabola is equidistant from the focus and the directrix.
Axis of Symmetry: - The line passing through the vertex and the focus, which divides the parabola into two symmetric parts. For the equations given, the axis is either the x-axis or the y-axis.
Vertex of the Parabola: - The vertex is the point where the parabola changes direction. It is the midpoint between the focus and the directrix. In the standard form of the parabola, the vertex is at (0, 0).
Latus Rectum: - The latus rectum is a line segment perpendicular to the axis of symmetry that passes through the focus. Its endpoints lie on the parabola, and its length is 4a . This is important in understanding the width of the parabola at the focus.
General Equation of a Parabola: - The general form of the equation of a parabola with vertex (h, k) is:
For a vertically oriented parabola:
(x - h)^2 = 4a(y - k) \)
For a horizontally oriented parabola:
(y - k)^2 = 4a(x - h) \)
Here, (h, k) is the vertex, and the value of a determines the shape and orientation of the parabola.
Applications of Parabolas: - Parabolas have practical applications in fields like physics and engineering. For example, satellite dishes and car headlights are designed with parabolic reflectors to focus light or signals.
Parametric Equations of a Parabola
Tangent and Normal to a Parabola
Intersection of a Line and a Parabola: - Solving the system of equations involving a line and a parabola helps in finding the points of intersection. This is useful for solving problems where tangents or chords are involved.
Parabola: - A parabola is a U-shaped curve that can open upward, downward, left, or right. It is defined as the set of all points equidistant from a fixed point called the focus and a fixed line called the directrix. The vertex is the point where the parabola changes direction, and the axis of symmetry is the line that divides the parabola into two mirror-image halves.
Key Features of a Parabola:
Vertex: - The point where the parabola changes direction.
Focus: - A fixed point inside the parabola used to define its shape.
Directrix: - A fixed line outside the parabola used to define its shape.
Axis of Symmetry: - The line that divides the parabola into two mirror-image halves.
Standard Equation of a Parabola
The general equation of a parabola
Parabola equation
Parabola formula
Parabola definition
Parabola examples
Parabola focus and directrix
Parabola graph
Parabola properties
Applications of parabola
Vertex form of parabola
Parabola in real life
Parabola explained
How to graph a parabola
Parabola derivation
Parabola focus and directrix explained
Real-life applications of parabolas
Parabola problems and solutions
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