Length Of The Tangent





In this page we are going to see how to find the length of the tangent to the circle from the point (x₁,y₁)

Length of the tangent = √ (x₁² + y₁² + 2gx₁ +2fy₁ + c)

(i) If the length is 0.Then we can say the given point must be on the circle.

(ii) If the length is > 0 .Then we can say the point must be outside the circle.

(iii) If the length is < 0 .Then we can say the point must be inside the circle.

Example 1:

Find the length of tangent to the circle x² + y² -4x - 3y + 12 = 0from the point (2,3)

Solution:

Length of the tangent = √ (x₁² + y₁² + 2gx₁ +2fy₁ + c)

Here x₁ = 2 and y₁ = 3

                   =  √ (2)² + 3² - 4(2) - 3(3) + 12

                   =  √ 4 + 9 - 8 - 9 + 12        

                   =  √4 + 12 - 8                    

                   =  √ 16 - 8        

                   =  √8                        

                   =  2√2 units       

Example 2:

Show that the point (2,-1) lies out side the circle x²+y²-6x-8y+12 = 0

Solution:

Length of the tangent = √ (x₁² + y₁² + 2gx₁ +2fy₁ + c)

Here x₁ = 2 and y₁ = -1

                   =  √ (2)² + (-1)² -6(2) -8 (-1) + 12

                   =  √ 4 + 1 - 16 + 8 + 12        

                   =  √5 + 8 + 12 + 1 - 8                    

                   =  √ 26 - 8        

                   =  √18                        

                   =  3√2 units       

The length of tangent is positive so the given point lies outside the circle.

                   

Example 3:

Find the length of tangent to the circle  x²+y²-4x+8y-5 = 0 from the point (2,1)

Solution:

Length of the tangent = √ (x₁² + y₁² + 2gx₁ +2fy₁ + c)

Here x₁ = 2 and y₁ = 1

                   =  √ 2² + 1² - 4(2) + 8(1) -5

                   =  √ 4 + 1 - 8 + 8 -5        

                   =  √5 -5                    

                   =  √0        

                   =  0                        

The length of tangent is zero so the given point lies on the circle.length of the tangent

Related Topics

Quote on Mathematics

“Mathematics, without this we can do nothing in our life. Each and everything around us is math.

Math is not only solving problems and finding solutions and it is also doing many things in our day to day life.  They are: 

It subtracts sadness and adds happiness in our life.    

It divides sorrow and multiplies forgiveness and love.

Some people would not be able accept that the subject Math is easy to understand. That is because; they are unable to realize how the life is complicated. The problems in the subject Math are easier to solve than the problems in our real life. When we people are able to solve all the problems in the complicated life, why can we not solve the simple math problems?

Many people think that the subject math is always complicated and it exists to make things from simple to complicate. But the real existence of the subject math is to make things from complicate to simple.”