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Equation of a Tangent

Equation of a Tangent

How do we find the equation of a tangent to a circle?

  • First, make sure you are familiar with Equations of Straight Lines and Perpendicular Gradients
  • A TANGENT just touches a circle (but does not cross it)
  • The gradient of the tangent at point P is PERPENDICULAR to the gradient of the radius OP
    • remember, perpendicular gradients multiply to -1 (they are negative reciprocals)
  • So if P is a point open parentheses x comma space y close parentheses on the circumference, then the gradient of the radius OP is fraction numerator y minus 0 over denominator x minus 0 end fraction equals y over x.
    • Therefore the gradient of the tangent to the circle at P is negative x over y
  • From here, use bold italic y bold equals bold italic m bold italic x bold plus bold italic c to find the equation of the tangent

Exam Tip

  • Solving simultaneous equations of circle and tangent only gives ONE SOLUTION
    • so if you are asked to show a line is a tangent, solve the simultaneous equations and show there is only one solution
  • ALWAYS draw a diagram to help!

Worked example

Circle Radius Tangent (before)Circles - Finding Tangents WE 1Circles - Finding Tangents WE 1a

Circles - Finding Tangents WE 2a

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