Question

Under which of the following conditions does a general second-degree equation ax² + 2hxy + by² +  2gx + 2fy + c = 0 (a ≠ 0) represent a circle?
1. h = g, a = b
2. h = g = f, a = b
3. h = 0, a = b
4. h = 0, g² + f² - c = a + b

Answer 1

Correct Answer - Option 3 : h = 0, a = b

Concept:

The general equation of any type of circle is represented by: x2 + y2 + 2gx + 2fy + c = 0, for all values of g, f and c.

Calculations:

Given, a general second-degree equation ax2 + 2hxy + by2 +  2gx + 2fy + c = 0 (a ≠ 0) represent a circle.

The general equation of any type of circle is represented by: x² + y² + 2gx + 2fy + c = 0, for all values of g, f and c.

⇒x2 + 2hxy + y2 + 2gx + 2fy + c = 0,

Comparing both equations, we get

⇒ a = 1, b = 1, 2h = 0

⇒ a = b and h = 0