# Tangent and Normal to a Parabola

### Tangent and Normal to a Parabola:

**1. Tangent to a Parabola:**

**Definition:**A tangent to a curve is a line that touches the curve at a single point without crossing through it.**Tangent Line Equation:**$y=mx+c$ or $x=ny+d$ at the point of tangency.**Slope:**At the point of tangency, the slope of the tangent is equal to the derivative of the parabola's equation.

**2. Normal to a Parabola:**

**Definition:**A normal to a curve at a particular point is a line perpendicular to the tangent at that point.**Normal Line Equation:**$y=-\frac{1}{m}x+e$ or $x=-\frac{1}{n}y+f$ at the point of tangency.**Slope:**At the point of tangency, the slope of the normal is the negative reciprocal of the slope of the tangent.

**3. Finding Tangents and Normals:**

**Vertical Parabola:**For $y=a{x}^{2}+bx+c$:- To find the tangent and normal at a specific point $({x}_{1},{y}_{1})$, differentiate the parabola equation and substitute ${x}_{1}$ to determine the slope.

**4. Key Properties:**

**Point of Tangency:**The point where the tangent touches the parabola.**Slope Relation:**The relationship between the slopes of the tangent and normal is that they are negative reciprocals of each other.

**5. Graphical Representation:**

- Plot the parabola and identify points where tangents or normals are required.
- Visualize the tangent touching the curve at a single point and the normal perpendicular to it.

### Example:

Given the equation of the parabola: $y={x}^{2}-4x+3$.

**1. Finding Tangent and Normal at a Specific Point:**

**a. Finding Tangent:**

Consider the point $P(2,-1)$ on the parabola.

**i. Deriving the Slope:**

- Differentiate the equation of the parabola $y={x}^{2}-4x+3$ to find the derivative.
- ${y}^{\mathrm{\prime}}=\frac{dy}{dx}=2x-4$
- At $x=2$, find the slope: ${y}^{\mathrm{\prime}}=2(2)-4=0$

**ii. Equation of Tangent:**

- At $x=2$ and $y=-1$, the tangent's equation is $y=0(x-2)-1$.
**Tangent Equation:**$y=-1$

**b. Finding Normal:**

**i. Slope of Normal:**

- The slope of the normal at $x=2$ is the negative reciprocal of the slope of the tangent.
- ${m}_{\text{normal}}=-\frac{1}{{m}_{\text{tangent}}}=-\frac{1}{0}$ (undefined)

**ii. Equation of Normal:**

- The equation of the normal line at $x=2$ and $y=-1$ is $x=2$ (vertical line).

### Analysis:

- At the point $P(2,-1)$ on the parabola $y={x}^{2}-4x+3$:
- The tangent is horizontal and intersects the point at $y=-1$.
- The normal is a vertical line passing through the point $x=2$.

### Graphical Representation:

- Plot the parabola $y={x}^{2}-4x+3$ on a graph.
- Identify the point $P(2,-1)$ on the curve.
- Visualize the tangent as a horizontal line and the normal as a vertical line passing through the point.