EDUCATIONAL GURU MATH: The tangent to a circle and touching circles

The tangent to a circle and touching circles

Finding the equation of a tangent to a circle

Finding the equation of the tangent to a circle is unlike finding the tangent to a polynomial. Because the tangent is a straight line, you need both a point and the gradient to find its equation. You are usually given the point - it's where the tangent meets the circle.

The method for finding the gradient uses the fact that the tangent is perpendicular to the radius at the point it meets the circle. Work out the gradient of the radius at the point the tangent meets the circle, and you can use the equation m_CP x m_tgt = -1 to find the gradient of the tangent.

Find the equation of the tangent to the circle

at the point P(5, -2) which lies on the circle.

centre = (1,1)
$m_{CP} = {{ - 2 - 1} \over {5 - 1}} = - {3 \over 4}$
hence $m_{tgt} = + {4 \over 3}$ since $m_{CP} \times m_{tgt} = - 1$

so equation of the tangent at P is

$\eqalign{ y - ( - 2) = {4 \over 3}(x - 5) \cr \cr 3(y + 2) = 4(x - 5) \cr \cr 3y - 4x + 26 = 0 \cr}$

Try this!
Find the equation of the tangent to the circle for the following examples.

Circle

at the point (5, 4)

centre = (1, 1): $m_{radius} = {{4 - 1} \over {5 - 1}} = {3 \over 4} \Rightarrow m_{tgt} = - {4 \over 3}$

$\eqalign{ y - 4 = - {4 \over 3}(x - 5) \cr \cr 3(y - 4) = - 4(x - 5) \cr \cr 3y - 12 = - 4x + 20 \cr \cr 3y + 4x = 32 \cr}$

is just as good.

Circle

at the point (2, 0)

centre $= (1, - {5 \over 2}):\;\;m_{radius} = {{0 - ( - {5 \over 2})} \over {2 - 1}} = {5 \over 2} \Rightarrow m_{tgt} = - {2 \over 5}$

$\eqalign{ y - 0 = - {2 \over 5}(x - 2) \cr \cr 5y = - 2(x - 2) \cr \cr 5y = - 2x + 4 \cr \cr 5y + 2x = 4 \cr}$

Touching circles

You may be asked to show that two circles are touching, and say whether they're touching internally or externally. To do this, you need to work out the radius and the centre of each circle. If the sum of the radii and the distance between the centres are equal, then the circles touch externally. If the difference between the radii and the distance between the centres are equal, then the circles touch internally.

Be smart!
This isn't a difficult thing to do, but many students forget this method - don't be one of them!

Do the circles

and

touch and if so, in what way?

$\eqalign{ C_1 = (1,1);C_2 = (5,4) \cr \cr C_1 C_2 = \sqrt {4^2 + 3^2 } = 5 \cr \cr r_1 = 3:r_2 = 2 \cr \cr \Rightarrow C_1 C_2 = r_1 + r_2 \cr}$

circles touch externally

Do the circles

and

touch and if so, in what way?

$\eqalign{ C_1 = (2,1);C_2 = (4,5) \cr \cr C_1 C_2 = \sqrt {2^2 + 4^2 } = \sqrt {20} = 2\sqrt 5 \cr \cr r_1 = \sqrt 5 :r_2 = \sqrt {45} = 3\sqrt 5 \cr \cr \Rightarrow C_1 C_2 = r_2 - r_1 \cr}$