Condition on Equation of Second Degree to Represent Circle

A general second-degree equation in two variables x and y can represent various conic sections, such as circles, ellipses, parabolas, or hyperbolas, depending on the conditions it satisfies. To specifically represent a circle, certain conditions must be met.

General Second-Degree Equation

The general form of a second-degree equation is:

$$ Ax^2 + Bxy + Cy^2 + Dx + Ey + F = 0 $$

Where A, B, C, D, E, and F are constants.

Circle Equation

A circle with center at $(h, k)$ and radius $r$ is represented by the equation:

$$ (x - h)^2 + (y - k)^2 = r^2 $$

Expanding this equation, we get:

$$ x^2 - 2hx + h^2 + y^2 - 2ky + k^2 - r^2 = 0 $$

Which simplifies to:

$$ x^2 + y^2 - 2hx - 2ky + (h^2 + k^2 - r^2) = 0 $$

Comparing this with the general second-degree equation, we can see that for a second-degree equation to represent a circle, it must satisfy certain conditions.

Conditions for a Second-Degree Equation to Represent a Circle

Here are the conditions that must be met:

1. The coefficients of $x^2$ and $y^2$ must be equal and non-zero (i.e., $A = C \neq 0$).
2. The coefficient of the $xy$ term must be zero (i.e., $B = 0$).
3. The equation must not represent a degenerate conic (i.e., it should not represent a point or a line).

Table of Differences and Important Points

Property	General Second-Degree Equation	Circle Equation
Coefficients of $x^2$ and $y^2$	May differ (A and C)	Must be equal and non-zero
Coefficient of $xy$	Can be non-zero (B)	Must be zero
Center	Not defined	$(h, k)$
Radius	Not defined	$r$
Degenerate Cases	May represent a point or line	Must represent a proper circle

Examples

Example 1: Identifying a Circle

Consider the equation:

$$ 2x^2 + 2y^2 - 8x + 4y + 4 = 0 $$

Let's check if it represents a circle:

1. Coefficients of $x^2$ and $y^2$: Both are 2 (equal and non-zero).
2. Coefficient of $xy$: There is no $xy$ term, so $B = 0$.
3. It does not represent a degenerate conic.

Since all conditions are met, this equation represents a circle.

Example 2: Not a Circle

Now consider the equation:

$$ x^2 + 2xy + y^2 + 3x + 4y + 5 = 0 $$

Checking the conditions:

1. Coefficients of $x^2$ and $y^2$: Both are 1 (equal and non-zero).
2. Coefficient of $xy$: It is 2, which is non-zero, so $B \neq 0$.

Since the coefficient of $xy$ is non-zero, this equation does not represent a circle.

Conclusion

For a second-degree equation to represent a circle, it must satisfy specific conditions regarding the coefficients of $x^2$, $y^2$, and $xy$. By comparing the given equation with the standard form of a circle's equation, one can determine whether it represents a circle or not. Understanding these conditions is crucial for solving problems related to circles in exams.