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=1mx+e or x=1ny+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=ax2+bx+c:
    • To find the tangent and normal at a specific point (x1,y1), differentiate the parabola equation and substitute x1 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=x24x+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=x24x+3 to find the derivative.
  • y=dydx=2x4
  • At x=2, find the slope: y=2(2)4=0

ii. Equation of Tangent:

  • At x=2 and y=1, the tangent's equation is y=0(x2)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.
  • mnormal=1mtangent=10 (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=x24x+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=x24x+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.