Circle: Standard and General Equations and Properties

Introduction to Circles

A circle is a fundamental geometric shape defined as the set of all points in a plane that are at a fixed distance from a fixed point. The fixed point is called the center of the circle, and the fixed distance is called the radius. Understanding the equations and properties of circles is crucial in coordinate geometry, as it allows us to describe and analyze their position, size, and relationships with other geometric figures.

Standard Equation of a Circle

The standard equation of a circle is derived using the distance formula. Let the center of the circle be denoted by $(h, k)$ and its radius be $r$. If $(x, y)$ is any point on the circle, then the distance between the center $(h, k)$ and the point $(x, y)$ is equal to the radius $r$. Using the distance formula, we have:

$\sqrt{(x-h)^2 + (y-k)^2} = r$

Squaring both sides, we get the standard equation of a circle:

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

This equation represents a circle with center $(h, k)$ and radius $r$. The values of $h$, $k$, and $r$ determine the specific position and size of the circle in the coordinate plane.

Special Case: Circle with Center at the Origin

If the center of the circle is at the origin $(0, 0)$, then $h=0$ and $k=0$. The standard equation simplifies to:

$x^2 + y^2 = r^2$

This equation represents a circle centered at the origin with radius $r$. This is a very common form and is frequently used in problems.

General Equation of a Circle

The general equation of a circle is obtained by expanding the standard equation and rearranging the terms. Expanding $(x-h)^2 + (y-k)^2 = r^2$, we get:

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

Rearranging the terms to match the general form:

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

This equation is typically written in the form:

$x^2 + y^2 + 2gx + 2fy + c = 0$

By comparing the two forms, we can relate the coefficients $g$, $f$, and $c$ to the center $(h, k)$ and radius $r$ of the circle:

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

Substituting $h = -g$ and $k = -f$ into the expression for $c$:

$c = (-g)^2 + (-f)^2 - r^2$

$c = g^2 + f^2 - r^2$

From this, we can find the radius $r$:

$r^2 = g^2 + f^2 - c$

$r = \sqrt{g^2 + f^2 - c}$

Therefore, for a circle represented by the general equation $x^2 + y^2 + 2gx + 2fy + c = 0$:

• The center is $(-g, -f)$.
• The radius is $\sqrt{g^2 + f^2 - c}$.

It is important to note that for the equation to represent a real circle, the radius must be a real and positive value. This means $g^2 + f^2 - c > 0$. If $g^2 + f^2 - c = 0$, the equation represents a point circle (a single point at the center). If $g^2 + f^2 - c < 0$, the equation represents an imaginary circle (no real points satisfy the equation).

Properties of Circles

Circles possess several important geometric properties that are useful in solving problems. These properties are often derived from the definition of a circle and its standard or general equations.

1. Diameter

A diameter of a circle is any straight line segment that passes through the center of the circle and whose endpoints lie on the circle. The length of the diameter is twice the length of the radius ($d = 2r$).

2. Chord

A chord is a straight line segment whose endpoints both lie on the circle. The diameter is the longest possible chord of a circle.

3. Tangent

A tangent to a circle is a line that touches the circle at exactly one point, called the point of tangency. The radius drawn to the point of tangency is perpendicular to the tangent line.

4. Secant

A secant is a line that intersects a circle at two distinct points.

5. Intercepts

A circle can intersect the coordinate axes. The points where the circle intersects the x-axis are called x-intercepts, and the points where it intersects the y-axis are called y-intercepts.

To find the x-intercepts, set $y=0$ in the circle's equation. For $x^2 + y^2 + 2gx + 2fy + c = 0$, setting $y=0$ gives:

$x^2 + 2gx + c = 0$

The solutions for $x$ are the x-coordinates of the x-intercepts. The number of real x-intercepts depends on the discriminant of this quadratic equation ($ (2g)^2 - 4(1)(c) = 4g^2 - 4c $). If $g^2 - c > 0$, there are two distinct x-intercepts. If $g^2 - c = 0$, there is one x-intercept (the circle is tangent to the x-axis). If $g^2 - c < 0$, there are no real x-intercepts.

Similarly, to find the y-intercepts, set $x=0$:

$y^2 + 2fy + c = 0$

The solutions for $y$ are the y-coordinates of the y-intercepts. The number of real y-intercepts depends on $f^2 - c$. If $f^2 - c > 0$, there are two distinct y-intercepts. If $f^2 - c = 0$, there is one y-intercept (the circle is tangent to the y-axis). If $f^2 - c < 0$, there are no real y-intercepts.

6. Position of a Point Relative to a Circle

For a circle defined by the equation $S = x^2 + y^2 + 2gx + 2fy + c = 0$, consider a point $P(x_1, y_1)$. We can determine the position of $P$ relative to the circle by substituting its coordinates into the expression $S_1 = x_1^2 + y_1^2 + 2gx_1 + 2fy_1 + c$:

• If $S_1 > 0$, the point $P$ lies outside the circle.
• If $S_1 = 0$, the point $P$ lies on the circle.
• If $S_1 < 0$, the point $P$ lies inside the circle.

This is because $S_1$ is related to the square of the distance from the point to the center compared to the square of the radius.

7. Family of Circles

A family of circles refers to a collection of circles that share a common property, such as passing through the intersection points of two given circles or being tangent to a given line at a given point.

Equation of a Circle in Different Forms

1. Standard Form (Center-Radius Form):

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

Center: $(h, k)$, Radius: $r$. This form is most useful when the center and radius are known or easily determined.

2. General Form:

$x^2 + y^2 + 2gx + 2fy + c = 0$

Center: $(-g, -f)$, Radius: $\sqrt{g^2 + f^2 - c}$. This form is useful when the equation is given in expanded form.

3. Diameter Form:

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

$(x - x_1)(x - x_2) + (y - y_1)(y - y_2) = 0$

This form is derived from the property that the angle subtended by a diameter at any point on the circumference is a right angle. The slopes of the lines joining $(x,y)$ to $(x_1, y_1)$ and $(x_2, y_2)$ are perpendicular.

4. Parametric Form:

For a circle with center $(h, k)$ and radius $r$, the parametric equations are:

$x = h + r \cos \theta$

$y = k + r \sin \theta$

where $\theta$ is the parameter, typically representing the angle made by the radius to the point $(x, y)$ with the positive x-axis (or a line parallel to it passing through the center). If the center is at the origin ($h=0, k=0$), the equations become $x = r \cos \theta$ and $y = r \sin \theta$. This form is useful for problems involving motion on a circular path or when dealing with angles.

Circle Passing Through Three Given Points

A unique circle can be drawn through any three non-collinear points. To find the equation of such a circle, we can use the general form $x^2 + y^2 + 2gx + 2fy + c = 0$. Substitute the coordinates of the three given points into this equation to obtain three linear equations in terms of $g$, $f$, and $c$. Solving these three equations will give the values of $g$, $f$, and $c$, and thus the equation of the circle.

Alternatively, we can find the center of the circle by finding the intersection of the perpendicular bisectors of the lines joining any two pairs of the given points. The distance from this center to any of the three points will be the radius.

Circle Touching Axes

A circle can touch one or both of the coordinate axes. The conditions for touching the axes are related to the relationship between the radius and the coordinates of the center.

• Circle touching the x-axis: The distance from the center $(h, k)$ to the x-axis is $|k|$. For the circle to touch the x-axis, this distance must be equal to the radius $r$. So, $r = |k|$. The equation becomes $(x-h)^2 + (y-k)^2 = k^2$.
• Circle touching the y-axis: The distance from the center $(h, k)$ to the y-axis is $|h|$. For the circle to touch the y-axis, this distance must be equal to the radius $r$. So, $r = |h|$. The equation becomes $(x-h)^2 + (y-k)^2 = h^2$.
• Circle touching both axes: For a circle to touch both axes, the distance from its center $(h, k)$ to both axes must be equal to its radius. This implies $r = |h| = |k|$.
• If the circle is in the first quadrant, its center is $(r, r)$, and the equation is $(x-r)^2 + (y-r)^2 = r^2$.
• If the circle is in the second quadrant, its center is $(-r, r)$, and the equation is $(x+r)^2 + (y-r)^2 = r^2$.
• If the circle is in the third quadrant, its center is $(-r, -r)$, and the equation is $(x+r)^2 + (y+r)^2 = r^2$.
• If the circle is in the fourth quadrant, its center is $(r, -r)$, and the equation is $(x-r)^2 + (y+r)^2 = r^2$.

Example Problem

Find the equation of the circle whose center is $(2, -3)$ and which passes through the point $(5, 1)$.

Solution:

The center of the circle is $(h, k) = (2, -3)$.

The radius $r$ is the distance between the center $(2, -3)$ and the point $(5, 1)$ on the circle.

Using the distance formula:

$r = \sqrt{(5-2)^2 + (1-(-3))^2}$

$r = \sqrt{(3)^2 + (4)^2}$

$r = \sqrt{9 + 16}$

$r = \sqrt{25}$

$r = 5$

Now, using the standard equation of a circle $(x-h)^2 + (y-k)^2 = r^2$:

$(x-2)^2 + (y-(-3))^2 = 5^2$

$(x-2)^2 + (y+3)^2 = 25$

Expanding this equation:

$x^2 - 4x + 4 + y^2 + 6y + 9 = 25$

$x^2 + y^2 - 4x + 6y + 13 = 25$

$x^2 + y^2 - 4x + 6y - 12 = 0$

Thus, the equation of the circle is $(x-2)^2 + (y+3)^2 = 25$ or $x^2 + y^2 - 4x + 6y - 12 = 0$.