Notes on Tangent of a Circle

Definition of Tangent

• A tangent is a straight line that touches a curve at exactly one point.
• It does not change regardless of the curve it touches.

Properties of Tangents in Geometry

• The tangent line is always perpendicular to the radius at the point of contact.
  • If OA is a radius, the tangent is always at 90 degrees to OA.

Analytical Geometry Concepts

• In analytical geometry, for two perpendicular lines:
  • The product of their gradients equals -1.
  • If gradient of radius is m1, then gradient of tangent (m2) is calculated as:
    • m1 * m2 = -1
    • m2 = -1/m1*

Equation of a Straight Line

• The general form of a straight line equation is:
  • y = mx + c
• Alternatively, it can also be expressed as:
  • y - y1 = m(x - x1)
• m represents the gradient of the line.

Finding the Tangent Equation to a Circle

1. Identify the center of the circle and its equation.
2. Determine the gradient of the radius (m1).
3. Use the relationship between gradients to find the gradient of the tangent (m2).
4. Substitute m2 and a point (x1, y1) into the equation of the straight line to find the tangent equation.

Example Problem

Problem Statement

• Find the equation of the tangent APB which touches a circle with center C and equation:
  x - 3² + y + 1² = 20
• Point P is (5, 3).

Solution Steps

1. Identify Center:
   • C = (3, -1)
2. Determine Gradient of PC:
   • Gradient of PC (m1) is calculated to be 2.
3. Find Gradient of Tangent (m2):
   • m2 = -1/m1 = -1/2.
4. Substitute into Line Equation:
   • Using the point P (5, 3):
   • Substitute to find the equation:
     • y = -1/2x + 11/2.

Conclusion

• The equation of the tangent is:
  y = -1/2x + 11/2
• This method outlines how to calculate the equation of a tangent to a circle.