Problem 29 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the circle. Identify its center and radius. $$x^{2}+8 x+y^{2}+2 y+8=0$$

Step-by-Step Solution

Answer

The center of the circle is at (-4, -1) and the radius of the circle is 3.

Step 1: Complete the square for x and y components in the circle equation

Rearrange the terms into groups and complete the square on the x-terms and y-terms. \[x^{2}+8x+y^{2}+2y=-8\] becomes \[(x^{2}+8x)+(y^{2}+2y)=-8\] which further simplifies to \[(x+4)^2 - 16 + (y+1)^2 - 1 = -8\]

Step 2: Simplify the equation

Simplify the equation to the standard form of a circle equation by moving the constants to the right-hand side. \[(x+4)^2 - 16 + (y+1)^2 - 1 = -8\] can be simplified to \[(x+4)^2 + (y+1)^2 = 9\].

Step 3: Identify the circle's parameters

The standard form of the equation is \((x-h)^2 + (y-k)^2 = r^2\). By comparing this to our simplified equation of \[(x+4)^2 + (y+1)^2 = 9\], we can identify \(h = -4\), \(k = -1\), and \(r = 3\).

Key Concepts

Completing the Square

Completing the square is a crucial technique used in algebra to transform quadratic equations into a simpler form. For a circle equation, it helps in identifying the center and the radius. Here is how you can complete the square:

• Take a quadratic expression like \(x^2 + 8x\) or \(y^2 + 2y\).
• For each quadratic expression, half the coefficient of the linear term, square it, and then add and subtract this value within the expression.

For example:

• For \(x^2 + 8x\), half of 8 is 4, and 4 squared is 16, so it becomes \((x+4)^2 - 16\).
• For \(y^2 + 2y\), half of 2 is 1, and 1 squared is 1, so it becomes \((y+1)^2 - 1\).

These steps convert the original equation into a format that reveals the circle’s center and radius clearly.

Standard Form of a Circle

The standard form of a circle's equation is essential to easily identify its geometrical properties. This form is \[(x-h)^2 + (y-k)^2 = r^2\]where the circle’s center is \((h, k)\) and \(r\) is the radius. After completing the square, the equation appeared as:

• \((x+4)^2 + (y+1)^2 = 9\)

This form is directly comparable with the standard circle equation to extract the properties of the circle readily.

Center of a Circle

The center of a circle is one of the key features that helps position it within the coordinate plane. Once you have your equation in the standard form, identifying the center \((h, k)\) becomes effortless.

• From the equation \((x+4)^2 + (y+1)^2 = 9\), compare it to \((x-h)^2 + (y-k)^2 = r^2\).
• The terms \(x+4\) and \(y+1\) reveal the center coordinates as \(h = -4\) and \(k = -1\).

Thus, the circle's center is at \((-4, -1)\), guiding you both in graphical plotting and analysis.

Radius of a Circle

The radius of a circle is another fundamental characteristic, determining its size and scale within the coordinate space. Once the equation is in standard form, extracting the radius is straightforward.

• The equation \((x+4)^2 + (y+1)^2 = 9\) shows the right side as \(9\).
• Since the standard form is \((x-h)^2 + (y-k)^2 = r^2\), you find \(r^2 = 9\).
• Solving for \(r\), you find that \(r = 3\).

Therefore, the radius of the circle is 3, which indicates its span from the center to any point on the circle's edge.