Problem 62 - Calculus for Biology and Medicine

Question

Find the center and the radius of the circle given by the equation \(x^{2}+y^{2}+2 x-4 y+1=0\). (To do this, you must complete the squares.)

Step-by-Step Solution

Answer

The center is (-1, 2) and the radius is 2.

Step 1: Rearrange the Equation

Start by grouping the terms of the equation by their respective variables: \[ x^{2} + 2x + y^{2} - 4y + 1 = 0 \] We'll complete the square for both \(x\) and \(y\) terms.

Step 2: Completing the Square for x

For the expression \( x^{2} + 2x \), we take the coefficient of \(x\), which is 2, divide it by 2, and square the result: \[ \left(\frac{2}{2}\right)^2 = 1 \] Add and subtract 1 to complete the square:\[ (x^{2} + 2x + 1 - 1) \] Simplifies to \( (x + 1)^2 - 1 \).

Step 3: Completing the Square for y

For the expression \( y^{2} - 4y \), take the coefficient of \(y\), which is -4, divide by 2, and square the result:\[ \left(\frac{-4}{2}\right)^2 = 4 \] Add and subtract 4 to complete the square:\[ (y^{2} - 4y + 4 - 4) \] Simplifies to \( (y - 2)^2 - 4 \).

Step 4: Rewrite the Equation

Substitute the completed squares back into the original equation:\[ (x + 1)^2 - 1 + (y - 2)^2 - 4 + 1 = 0 \] Combine and simplify the constant terms:\[ (x + 1)^2 + (y - 2)^2 - 4 = 0 \] thus:\[ (x + 1)^2 + (y - 2)^2 = 4 \]

Step 5: Identify the Center and Radius

The rewritten equation \( (x + 1)^2 + (y - 2)^2 = 4 \) is in the standard circle form \( (x - h)^2 + (y - k)^2 = r^2 \). Here, \( h = -1 \) and \( k = 2 \), and \( r^2 = 4 \) which means \( r = 2 \).Thus, the center of the circle is \((-1, 2)\) and the radius is \(2\).

Key Concepts

Circle Equation

A circle equation in coordinate geometry describes all the points (x, y) that lie on the circle with respect to its center point. Any such equation can initially appear cluttered or more complex in non-standard form. In many exercises, as in our example, you may have terms involving both x and y that do not initially present the equation in a recognizable form. This is where completing the square becomes crucial.

• To identify a circle equation, look for terms involving both variables, often presented as squared terms combined with linear terms.
• The primary goal is to manipulate or simplify the equation into a form that easily reveals the center and radius of the circle.

By completing the square for both x and y, these equations get restructured to align with the standard form, simplifying the process of identifying the circle's critical attributes.

Standard Form of a Circle

The standard form of a circle in mathematics is crucial for identifying the circle's geometric properties directly from its equation. This standard form is expressed as: \[(x - h)^2 + (y - k)^2 = r^2\] Here, (h, k) represents the circle's center, and r is the radius.

• This format neatly separates the circle's variables, showing them as perfect squares deriving from their respective centers.
• It allows quick visualization: from the equation alone, you can readily identify both the center and the radius.

In our earlier exercise, transforming the initial messy equation into the standard form required completing the square for both the x and y sections.
This reveals the circle's true structure, making it simpler to interpret and calculate subsequent geometry problems.

Center and Radius of a Circle

The center and radius are fundamental properties that define a circle. In the standard equation \[(x - h)^2 + (y - k)^2 = r^2\], these parameters tell you everything about the circle's size and position.

• The center, given by the coordinates (h, k), is the fixed point equidistant from all points on the circle.
• The radius, given by r, represents the constant distance from the center to any point on the perimeter of the circle.

From our example, completing the square allowed us to convert the original equation into the form where these properties were explicit.
The resulting equation indicated a center at (-1, 2) and a radius of 2. Here, the process of completing the square streamlined the identification of these crucial circle attributes, ensuring accuracy and ease in solving or graphing related problems.