# Parabola

1. Definition of a Parabola:

• A parabola is a U-shaped curve formed by intersecting a cone with a plane parallel to its side.
• It is defined as the set of points that are equidistant from a fixed point (the focus) and a fixed line (the directrix).

2. Equation of a Parabola:

• General equation: $y=a{x}^{2}+bx+c$ or $x=a{y}^{2}+by+c$, where $a$, $b$, and $c$ are constants..
• Standard form for a vertical parabola with its vertex at $\left(h,k\right)$: $y=a\left(x-h{\right)}^{2}+k$.
• Standard form for a vertical axis of symmetry: $y=a{x}^{2}+bx+c$.
• Standard form for a horizontal axis of symmetry: $x=a{y}^{2}+by+c$.
• For a horizontal parabola, the equation becomes $x=a{y}^{2}+by+c$ or $x=a\left(y-k{\right)}^{2}+h$.

3. Elements of a Parabola:

• Focus and Directrix:
• Focus: A fixed point on the axis of symmetry, equidistant from any point on the parabola.
• Directrix: A fixed line perpendicular to the axis of symmetry, also equidistant from any point on the parabola.
• Axis of Symmetry:
• A line that divides the parabola into two symmetric halves. For a vertical parabola, the axis is parallel to the y-axis; for a horizontal parabola, it's parallel to the x-axis.
• Vertex:
• The point where the parabola intersects its axis of symmetry. It's the minimum (for a downward-opening parabola) or maximum (for an upward-opening parabola) point on the graph.

4. Properties of Parabolas:

• The parabola opens upward if the coefficient $a$ is positive and downward if $a$ is negative in the equation $y=a{x}^{2}+bx+c$.
• If the coefficient of the squared term is $1$ (i.e., $y={x}^{2}$), the parabola is symmetric around the y-axis.
• If the squared term is ${x}^{2}$ in $x=a{y}^{2}+by+c$, the parabola is symmetric around the x-axis.

5. Types of Parabolas:

• Vertical Parabola:
• Opens either upward (if $a>0$) or downward (if $a<0$).
• Standard form: $y=a{x}^{2}+bx+c$.
• Horizontal Parabola:
• Opens either rightward (if $a>0$) or leftward (if $a<0$).
• Standard form: $x=a{y}^{2}+by+c$.

Applications of Parabolas:

• Real-life applications include projectile motion (like the path of a thrown object), satellite dish designs, and reflecting telescopes.
• Used in engineering for designing lenses and mirrors to focus light and in architecture for designing arches.