## 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

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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: