# 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 $\left({x}_{1},{y}_{1}\right)$, 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 $P\left(2,-1\right)$ 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: ${y}^{\mathrm{\prime }}=2\left(2\right)-4=0$

ii. Equation of Tangent:

• At $x=2$ and $y=-1$, the tangent's equation is $y=0\left(x-2\right)-1$.
• 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 $P\left(2,-1\right)$ 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 $P\left(2,-1\right)$ on the curve.
• Visualize the tangent as a horizontal line and the normal as a vertical line passing through the point.