# A circle touches two perpendicular lines #2x-3y=15# and #3x+2y=3# at the points #A(6,-1)#, #B(1,0)# respectively. Find the equation of the circle?

##### 2 Answers

#### Explanation:

The tangent to a circle

at the point

Thus, the tangent at

or

Since this is the line

Again, the tangent at

or

Since this is the line

Hence

Thus

So,

and

Thus the circle is

#### Explanation:

The following is a graph of:

Because the center must be located on lines that are perpendicular to the points of tangency, we can use equations [1] and [2] to write two equations that must intersect at the center.

Set equation [1] equal to an arbitrary constant,

Set equation [2] equal to an arbitrary constant

Substitute point B into equation [1.1] and solve for

Substitute the above into equation [1.1]:

Substitute point A into equation [2.1] and solve for

Substitute the above into equation [2.1]:

Add equations [1.2] and [2.2] to the graph:

Please understand that the center of the circle must be at the intersection of equations [1.2] and [2.2], therefore, we shall solve them as a system of equations:

The center of the circle is at the point

The standard Cartesian equation of a circle is:

where

Substitute point C into equation [3]:

To find the radius, we shall substitute point B into equation [3.1]:

Substitute the above into equation [3.1]:

Add equation [3.2] to the graph:

Please observe that equation [3.2] touches [1] and [2] and points A and B respectively, therefore, equation [3.2] is the correct equation of the circle.