Tangents and normals: parametric type

Tangents and Normals: Parametric Type

Understanding Tangents and Normals

  • A tangent line to a curve at a particular point is a straight line that just touches the curve at that point.
  • A normal to a curve at a given point is a straight line that is perpendicular to the tangent at that point.
  • Both the tangent and normal are defined at a point on a curve.

Parametric Equations

  • Parametric equations are a set of equations that define a group of quantities as functions of a variable, the parameter.
  • The parameter normally represents the time factor, such as in physics or engineering.
  • When dealing with parametric equations, we often have to convert between parametric form and Cartesian form, where y = f(x).

Tangents and Normals in Parametric Terms

  • The slope of a tangent to a curve at a certain point in parametric terms is given by dy/dx at that point. This is determined by taking dy/dt ÷ dx/dt, where t is the parameter.
  • To find the equation of the tangent line, we use the point-slope form of a linear equation: y - y1 = m(x - x1), where m is the slope of the tangent and (x1, y1) is a point on the curve.
  • The slope of the normal line to a curve at a certain point is the negative reciprocal of the slope of the tangent at that point. If the slope of the tangent is m, then the slope of the normal is -1/m.
  • The equation of a normal to a curve at a point in parametric terms can similarly be found by using the point-slope form of a linear equation and the negative reciprocal of the slope of the tangent.

Procedures for Tangents and Normals to Parametric Curves

  • To obtain the tangent or normal to a curve in parametric form at a particular point, we need to do the following:
    • Substitute the value of the parameter into both x and y parametric equations to get the coordinates of the point.
    • Find dy/dx, the derivative of y with respect to x (using dy/dt ÷ dx/dt), and evaluate it at that point to find the gradient of the tangent.
    • Use the point-slope form of the linear equation to obtain the equation of the tangent.
    • Use the negative reciprocal of the tangent’s gradient to find the gradient of the normal.
    • Using the same point and the new gradient, use the point-slope form of a linear equation to find the equation of the normal.

These steps and principles are instrumental in understanding and solving problems about tangents and normals in the parametric form. This will form a pivotal part of further mathematical studies.