Question

Write the standard form of the equation and the general form of the equation of each circle of radius \(r\) and center \((h, k)\). Graph each circle. $$ r=4 ;(h, k)=(2,-3) $$

Short answer

Standard form: \((x - 2)^2 + (y + 3)^2 = 16\); General form: \(x^2 + y^2 - 4x + 6y - 3 = 0\).

Step-by-step solution

Identify the Given Information

The radius \(r\) is given as 4, and the center of the circle \((h, k)\) is given as \((2, -3)\).\

Write the Standard Form of the Circle

The standard form of the equation of a circle with radius \(r\) and center \((h, k)\) is \((x - h)^2 + (y - k)^2 = r^2\).Substituting the given values: \((x - 2)^2 + (y + 3)^2 = 4^2\).This simplifies to: \((x - 2)^2 + (y + 3)^2 = 16\).\

Expand to Find the General Form

To convert from standard form to general form, expand the equation \((x - 2)^2 + (y + 3)^2 = 16\).First, expand both squared terms: \(x^2 - 4x + 4 + y^2 + 6y + 9 = 16\).Next, combine like terms: \(x^2 + y^2 - 4x + 6y + 13 = 16\).Subtract 16 from both sides to set the equation to 0: \(x^2 + y^2 - 4x + 6y - 3 = 0\).\

Graph the Circle

To graph the circle, plot the center at the point \((2, -3)\) on the coordinate plane. Then, use the radius of 4 to draw a circle that extends 4 units in all directions from the center. The circle should intersect points \((6, -3)\), \((2, 1)\), \((-2, -3)\), and \((2, -7)\).\

Key concepts

standard form of a circle

The standard form of a circle's equation is a compact and convenient way to express all the necessary information about a circle. It specifies the circle's center and radius clearly. The standard form equation of a circle with center \( (h, k) \) and radius \( r \) is:
\((x - h)^2 + (y - k)^2 = r^2\).
This equation simply states that any point \( (x, y) \) on the circle is exactly \( r \) units away from the center \( (h, k) \). By substituting the given radius \( r = 4 \) and center \( (h, k) = (2, -3) \) into the standard form, we get:
\((x - 2)^2 + (y + 3)^2 = 16\).
This describes a circle centered at \( (2, -3) \) with a radius of 4 units.

general form of a circle

The general form of a circle's equation is less compact, but still valuable for understanding and solving equations involving circles. It looks like this:
\(Ax^2 + Ay^2 + Dx + Ey + F = 0\).
To convert the standard form to the general form, we need to expand and combine like terms. For our circle:
Start with the standard form: \((x - 2)^2 + (y + 3)^2 = 16\).
Expand:
\(x^2 - 4x + 4 + y^2 + 6y + 9 = 16\).
Combine like terms and rearrange:
\(x^2 + y^2 - 4x + 6y + 13 = 16\).
Subtract 16 from both sides to set the equation to 0:
\((x^2 + y^2 - 4x + 6y - 3 = 0)\).
This is now in the general form, where \( A = 1 \), \( D = -4 \), \( E = 6 \), and \( F = -3 \).

graphing a circle

Graphing a circle involves plotting its center and drawing a curve that represents all the points at a fixed distance (radius) from that center. Here’s how to graph our example circle.
First, locate the circle's center \( (2, -3) \) on the coordinate plane. Mark this point clearly. Next, using a compass or a steady hand, draw a circle that extends 4 units (the radius) in every direction from the center. This ensures the circle passes through the points \( (6, -3) \), \( (2, 1) \), \( (-2, -3) \), and \( (2, -7) \).
These points are found by moving horizontally and vertically from the center by the radius length. Ensure your circle is smooth and round, intersecting these points precisely.

radius and center

The radius and center of a circle are fundamental in defining and understanding its equation and geometric properties. The center \( (h, k) \) is the point equidistant from every point on the circle. The radius \( r \) is the distance from this center to any point on the circle. For our example, the center is \( (2, -3) \) and the radius is 4 units.
To visualise, if you were to walk 4 units in any direction from the point \( (2, -3) \), you'd land exactly on the circumference of the circle. This relationship is the basis for the standard equation \( (x - h)^2 + (y - k)^2 = r^2 \). Knowing these values helps in both graphing the circle and converting between the standard and general forms of the circle's equation.