Externally Tangent Circles

Two intersect circles in a single point is known as tangent circles. It can be divided into two types of tangency: internal and external. By using the tangent circles many problems and constructions in geometry are solved. This type of problems have real-life applications. The followings are some real-time applications: trilateration and maximizing the use of materials.

                                             

I like to share this Tangent geometry with you all through my article. 

Illustration:

A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called Common External Tangent. A common tangent can be any one of the following:External tangent or Internal tangent.

                                                   
          From the above figure, we can see that line PQ is the common external tangent.

                                               
          From the above figure, we can see that line AB is the common external tangent.

Theorem for Externally Tangent Circles:

Theorem:If a line is tangent to a circle, it is perpendicular to the radius drawn to the point of tangency.
                                                       
Solution:
            If:        Line  AB is a tangent
                       D is point of tangent.
            Then:   OD ┴ AB

Example problems:

Example 1: Find the common external tangent from the below figure.

                                             
Solution:
            A tangent that is common to two circles and does not intersect the segment joining the centers of the circles is called common external tangent.
            From the given figure, the line CD touches both the circles on the same side of the line is the external tangent to the circle.
            Therefore, CD is the common external tangent line.

Example 2: Find the common external tangent from the below figure.
                                                                                    
Solution:
            The given figure is concentric circles. Concentric circles having the same center and not tangent.

Example 3: Find the common external tangent from the below figure.
                                                     
Solution:
          From the above figure, we can identify that given is two complete separate circles.
          The given figure have two external tangents.

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Example 4: Find the equation for tangent lines y=3x³ from the point x= 2.
Solution :
Given, y=3x³………….(1)
Step 1: To find the first derivative of y

                y ' = 3x ² ………..(2)
Step  2: To plug x =2 into y ' to find the slope at x.
                y ' = 3(2) ²
                y= 12
               Slope of tangent line =12
Step 3:To plug x=2 into y to find the y value of the tangent point
              y =3(2)³
              y=24 
              Hence tangent point (2,24)
Step 4: To plug the slope=12 and point (2,24) using the point-slope equation to find the equation     for the tangent line.
              y- y₁ =m (x-x₁)
              y- 24 =12 (x-2)
              y =12x- 24 + 24
              y=12x
Example 5: Find the slope of tangent line which is passing through the point (2, 0) and the curve y = 2x – x³
Solution:
            Step 1: Find the derivative y¹
                                y = 2x – x³
                                y¹ = 2 -3x²
            Step 2: To find the slope of tangent line,
                         Given point (2, 0) --- > (x, y)
                         y¹ = 2 -3(2)² ---- > 2 – 3(4) --- > 2 -12 = -10
                         Slope of the tangent line = -10         

Practice problems:

1. Determine the equation of the tangent line passing through the point (− 1, − 2) and having slope4 /7
Answer:   4x − 7y − 10 = 0
2. Determine the equation of the line with the slope 3 and y-intercept 4.
Answer: y = 3x + 4