Inside outside or on the circle

INSIDE OUTSIDE OR ON THE 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 :

Determine whether the points (− 2, 1), (0, 0) and (4, − 3) lie outside, on or inside the circle x² + y² − 5x + 2y − 5 = 0

Solution :

(i) First let us check where does the point (-2, 1) lie on the circle x² + y² − 5x + 2y − 5 = 0

PT = √(x₁² + y₁² + 2gx₁ +2fy₁ + c)

here x₁ = -2 and y₁ = 1

PT = √((-2)² + 1² − 5(-2) + 2(1) − 5)

PT² = (4 + 1 + 10 + 2 − 5)

PT²= (17 − 5)

PT²= 12 > 0

Hence the point (-2, 1) lies outside the circle.

(ii) (0, 0)

here x₁ = 0 and y₁ = 0

PT = √((0)² + 0² − 5(0) + 2(0) − 5)

PT² = (0 + 0 − 0 + 2 − 5)

PT²= -3 < 0

Hence the point (0, 0) lies inside the circle.

(ii) (4, -3)

here x₁ = 4 and y₁ = -3

PT = √(4² + (-3)² − 5(4) + 2(-3) − 5)

PT² = (16 + 9 − 20 - 6 − 5)

PT²= 25 - 20 - 6 - 5

= 20 - 31

= -11 < 0

Hence the point (4, -3) lies inside the circle.

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