Equation of tangent to a circle at a given point

Explanation

In order to find the equation of a tangent to a circle at a given point on it, there are two things which you require - the coordinates of the point of contact and slope of the tangent. Assuming that the coordinates of the point of contact are already given in the question, we just need to find the slope of the tangent to find its equation. For that, we will use a simple theorem, which says that the radius of a circle is always perpendicular to the tangent at the point of contact. Thus if we calculate the slope of the radius, we can calculate the slope of the tangent using the perpendicular slopes theorem. After getting the slope of the tangent, all we need to do is apply the point slope form to get the equation of the tangent.

Steps 

The above method can be outlined in the following steps:

1. Draw a diagram
2. Calculate slope of the radius
3. Calculate slope of the tangent
4. Form the equation of tangent

Example

An example will make it clearer.

Find the equation of the tangent to the circle `x^2 + y^2 = 25` at the point of contact `(4, 3)`.

We will follow the steps outlined above to solve this problem:

1. Draw a diagram to make the problem more clear

The radius is perpendicular to the tangent

The above diagram serves the purpose of making clear that the radius is perpendicular to the tangent, and so, knowing the slope of the radius, we will be able to find the slope of the tangent.

2.  Calculate slope of the radius

Slope of the radius in the above diagram can be calculated by using the slope formula. The radius connects the center of the circle with the point of contact. The center of the circle `x^2 + y^2 = 25` is (0, 0) and the point of contact is given (4, 3). Then, using the slope formula, the slope of the radius is

`m = (y_2 - y_1)/(x_2 - x_1) = (4 - 0)/(3 - 0) = 4/3`

3. Calculate the slope of the tangent

Since the tangent is perpendicular to the radius, hence its slope can be calculated by using the perpendicular slopes theorem. That is,

Slope of tangent * Slope of radius = -1

Let the slope of the tangent be 'm', then,

`m * 4/3 = -1`

Solving for 'm', we get

`m = -3/4`

4. Calculate the equation of the tangent using point slope form

Since we know the coordinates of a point (the point of contact) on the tangent and its slope, we can use the point slope form to get its equation.

`y - y_1 = m(x - x_1)`

`y - 3 = -3/4(x - 4)`

Solving for 'y',

`y = -3/4x + 6`

Thus, the equation of the tangent to the circle `x^2 + y^2 = 25` at the point (4, 3) is `y = -3/4x + 6`.

Hope the above example is clear. Thanks for reading the article. Please comment below.