Equation-Of-A-Tangent Aqa Higher

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Exam code:8300

Equation of a tangent

First, make sure you are familiar with equations of straight lines and perpendicular lines
A tangent just touches a circle (but does not cross it)
The tangent at point P is perpendicular to the radius OP
- remember, the gradients of perpendicular lines multiply to -1
  - they are negative reciprocals
So if P is a point $open parentheses a comma space b close parentheses$ on the circumference, then the gradient of the radius OP is $fraction numerator b minus 0 over denominator a minus 0 end fraction equals b over a$ .
- Therefore the gradient of the tangent to the circle at P is $negative a over b$
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

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!