Equation of Circle - Standard and General Forms

Equation of Circle - Standard and General Forms

Hey JEE aspirants! Circles are fundamental to coordinate geometry and appear frequently in JEE Main. Mastering the equation of a circle, in its various forms, is crucial for solving a wide range of problems. Let's dive in!

Standard Form of the Circle

The standard form of a circle's equation is derived directly from the definition of a circle: the set of all points equidistant from a center. Imagine a point $(x, y)$ moving in such a way that its distance from a fixed point $(h, k)$ is always equal to $r$. Using the distance formula, we get:

Squaring both sides to remove the square root gives us the standard form:

Here, $(h, k)$ represents the center of the circle, and $r$ is the radius. This form is incredibly useful because it immediately tells you the center and radius of the circle. For example, the equation $(x - 2)^2 + (y + 3)^2 = 16$ represents a circle with center $(2, -3)$ and radius $\sqrt{16} = 4$. Geometrically, this means the circle is centered at the point (2,-3) on the Cartesian plane, and every point on the circle is exactly 4 units away from this center.

General Form of the Circle

The general form of a circle's equation is obtained by expanding the standard form. Let's start with the standard form:

Expanding this, we get:

Rearranging the terms, we get:

Now, let's make the following substitutions:

• $g = -h$
• $f = -k$
• $c = h^2 + k^2 - r^2$

Substituting these into the equation, we obtain the general form:

While this form doesn't immediately reveal the center and radius, it's important for recognizing circles in more complex equations. Notice that the coefficients of $x^2$ and $y^2$ are both 1, and there is no $xy$ term. This is a key characteristic of a circle's equation. However, not every equation of this form represents a real circle. The radius must be a real number, implying $g^2 + f^2 - c > 0$.

Finding Center and Radius from General Form

Given the general form $x^2 + y^2 + 2gx + 2fy + c = 0$, we can determine the center and radius by comparing it to the expanded standard form. From our substitutions above, we know:

• $h = -g$
• $k = -f$
• $r^2 = g^2 + f^2 - c$

Therefore, the center and radius are given by:

For instance, consider the equation $x^2 + y^2 + 4x - 6y - 12 = 0$. Here, $2g = 4$, $2f = -6$, and $c = -12$. Thus, $g = 2$, $f = -3$, and the center is $(-2, 3)$. The radius is $\sqrt{2^2 + (-3)^2 - (-12)} = \sqrt{4 + 9 + 12} = \sqrt{25} = 5$. To find the center geometrically, complete the square in both $x$ and $y$. This allows you to rewrite the equation in standard form.

Circle Through Three Points

A unique circle can be defined by three non-collinear points. One way to find the equation of this circle is to assume the general form $x^2 + y^2 + 2gx + 2fy + c = 0$. If the circle passes through points $(x_1, y_1)$, $(x_2, y_2)$, and $(x_3, y_3)$, then these points must satisfy the equation:

This gives us a system of three linear equations in three unknowns ($g$, $f$, and $c$). Solve this system to find the values of $g$, $f$, and $c$, which will define the equation of the circle. Geometrically, the center of this circle is the point of intersection of the perpendicular bisectors of the lines joining the three points.

Parametric Form of Circle

The parametric form of a circle's equation provides a way to represent the coordinates of any point on the circle using a single parameter, usually denoted as $\theta$. Starting from the standard form $(x - h)^2 + (y - k)^2 = r^2$, we can express $x$ and $y$ in terms of trigonometric functions:

Here, $\theta$ is the angle that the radius to the point $(x, y)$ makes with the positive x-axis. As $\theta$ varies from $0$ to $2\pi$, the point $(x, y)$ traces out the entire circle. Geometrically, $r\cos\theta$ and $r\sin\theta$ are the horizontal and vertical components of the radius vector, respectively. The parametric form is particularly useful when dealing with problems involving the angle subtended by a chord at the center.

For example, the parametric equations for a circle centered at $(1, 2)$ with radius $3$ would be $x = 1 + 3\cos\theta$ and $y = 2 + 3\sin\theta$.

Diameter Form of Circle

If $(x_1, y_1)$ and $(x_2, y_2)$ are the endpoints of a diameter of a circle, then the equation of the circle can be directly written as:

This equation represents the circle whose diameter has endpoints $(x_1, y_1)$ and $(x_2, y_2)$. Geometrically, this equation is derived from the fact that the angle in a semicircle is a right angle. If $(x, y)$ is any point on the circle, then the lines joining $(x, y)$ to $(x_1, y_1)$ and $(x_2, y_2)$ are perpendicular, and the product of their slopes is -1.