Question

Which of the following points lies outside the circle x² + y² - 2x + 6y + 1 = 0?
1. (-1, -5)
2. (1, -5)
3. (-2, -6)
4. (2, -5)

Answer 1

Correct Answer - Option 3 : (-2, -6)

Concept:

If substituting the values (x₁, y₁) for x and y in the circle's equation x² + y² + 2gx + 2fy + c = 0, we get:

• x₁² + y₁² + 2gx₁ + 2fy₁ + c = 0, then (a, b) is on the circle.
• x12 + y12 + 2gx1 + 2fy1 + c < 0, then (a, b) is inside the circle.
• x12 + y12 + 2gx1 + 2fy1 + c > 0, then (a, b) is outside the circle.

 

Calculation:

Substituting the points in the equation of the circle, we get:

1) (-1, -5): (-1)2 + (-5)2 - 2(-1) + 6(-5) + 1 = 1 + 25 + 2 - 30 + 1 = -1.

2) (1, -5): (1)2 + (-5)2 - 2(1) + 6(-5) + 1 = 1 + 25 - 2 - 30 + 1 = -5.

3) (-2, -6): (-2)2 + (-6)2 - 2(-2) + 6(-6) + 1 = 4 + 36 + 4 - 36 + 1 = 9.

4) (2, -5): (2)2 + (-5)2 - 2(2) + 6(-5) + 1 = 4 + 25 - 4 - 30 + 1 = -4.

Hence, the correct answer is 3, (-2, -6) is outside the circle.