HOW TO DETERMINE IF A POINT IS INSIDE OR OUTSIDE A CIRCLE

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

To check if the given point lie on the circle, inside the circle or out side the circle, we use the formula for length of tangent.

Example 1 :

Show that the point (2, 3) lies inside the circle x² + y² − 6x − 8y + 12 = 0.

Solution :

The length of the tangent PT from P(x₁, y₁) to the circle

x² + y² + 2gx +2fy + c = 0 is

PT  =  x₁² + y₁² + 2gx₁ + 2fy₁ + c

PT  =  √(22 + 32 − 6.2 − 8.3 + 12)

  =  √(4 + 9 − 12 − 24 + 12)

  = − 11 < 0

The point (2, 3) lies inside the circle

Example 2 :

Show that the point (2,-1) lies outside 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 :

Is the point (7, − 11) lie inside or outside the circle x² + y² − 10x = 0 ?

Solution :

To know that where does the given point lie in the circle, we have to find the length of tangent. 

Length of the tangent

= √ (x₁² + y₁² + 2gx₁ +2fy₁ + c)

Here x₁ = 7 and y₁ = -11

  =  √ x² + y² − 10x

  =  √7² + (-11)² − 10(7)

  =  √49 + 121 - 70

  =  √ 170 - 70

  =  √100 =  10 > 0

Hence the given point (7, -11) outside the circle.

Related pages

• How to find the length of tangent
• How to find equation of tangent at the given point
• Inside outside or on the circle

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