## About "How to determine if a point is inside or outside a circle"

How to determine if a point is inside or outside a circle ?

Here we are going to see 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.

## How to verify that points lie on a circle ?

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

√ (x₁² + y₁² + 2gx₁ +2fy₁ + c)  =  0

## How to verify that points lie outside the circle ?

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

√ (x₁² + y₁² + 2gx₁ +2fy₁ + c)  >  0

## How to verify that points lie inside the circle ?

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

√ (x₁² + y₁² + 2gx₁ +2fy₁ + c)  <  0

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  =  x1+ 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.

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