# 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 $(h,k)$: $y=a(x-h{)}^{2}+k$.
- 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 $x=a(y-k{)}^{2}+h$.

**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.