## Equation of Tangent to a Circle

We can find the equation of a tangent to a circle in the following way.
Suppose we have a point
$(x_1,y_1)$
on a circle and a tangent at that point. The centre of the circle is at the point
$(x_{CENTRE},y_{CENTRE})$
$m_{RADIUS}=\frac{y_1-y_{CENTRE}}{x_1-x_{CENTRE}}$
and the tangent, being at right angles to the radius at the point where they meet on the circle, has gradient
$m_{TANGENT}=- \frac{1}{m_{RADIUS}}$
.
The equation of the tangent is then
$y-y_1 =m_{TANGENT}(x-x_1)$

Example: A circle has centre
$(3,2)$
. Find the equation of the tangent to the circle at the point
$(6,6)$
.
$(3,2)$
to
$(6,6)$
is
$m_{RADIUS}=\frac{6-2}{6-3}= \frac{4}{3}$
.
The gradient of the tangent is then
$m_{TANGENT} = -\frac{1}{4/3}=- \frac{3}{4}$
.
The equation of the tangent is
$y-2=- \frac{3}{4} (x-3)=- \frac{3}{4} x+\frac{9}{4}$
.
Hence
$y=-\frac{3}{4}x +\frac{9}{4}+2=-\frac{3}{4}x +\frac{17}{4}$