Problem 62 - Algebra 2

Question

Find the center and the radius of each circle. $$ x^{2}+2 x+1+y^{2}=4 $$

Step-by-Step Solution

Answer

Thus the center of the circle is \(-1, 0\) and the radius is \(\sqrt{3}\).

Step 1: Rewrite the Equation

We need to rewrite the given equation in the standard form of a circle to be able to identify the center and radius. In order to do this, we need to complete the square for both variables x and y separately. The equation is \(x^{2}+2 x+1+y^{2}=4\). First, group the x terms together and re-arrange the equation to: \(x^{2}+2x +y^{2}=4-1\) which simplifies to \(x^{2}+2x+y^{2}=3\).

Step 2: Recognise Pattern

Now we are looking to rewrite the x terms in the form of \((x-h)^{2}\), where h is the x-coordinate of the center. Notice that \((x+1)^{2}\) would simplify to \(x^{2}+2x+1\), which are the x terms in the equation. Therefore, we could rewrite the equation as \((x+1)^{2}+y^{2}=3\).

Step 3: Identify The Center and Radius

Now, from the equation \((x+1)^{2} + y^{2} = 3\) which is in the form of \((x-h)^{2} + (y-k)^{2} = r^{2}\), it is easy to identify the center and the radius. By comparing the two forms, we find that \(h = -1\) and \(k = 0\), so the center of the circle is \((-1, 0)\). Additionally, we find that \(r^{2} = 3\), so the radius of the circle is \(\sqrt{3}\).

Key Concepts

Completing the Square

Completing the square is a powerful technique used to transform a quadratic expression into a perfect square trinomial. This method is especially useful in circle geometry when rewriting equations to find the center and radius of a circle.

Here’s a quick guide on how to complete the square:

• Identify the quadratic and linear terms in the expression you wish to transform.
• For a term like \(x^2 + bx\), calculate \(\left(\frac{b}{2}\right)^2\).
• Add and subtract this value inside the expression to rewrite it as a perfect square trinomial.

For instance, in the problem at hand with terms \(x^2 + 2x\), we calculate \(\left(\frac{2}{2}\right)^2 = 1\). By adding and subtracting 1, we rewrite the expression as \((x+1)^2\).

This transformation simplifies analyzing quadratic expressions and makes certain properties, such as the vertex form, evident. In circle geometry, completing the square is instrumental in converting general quadratic forms into the recognizable standard form of a circle.

Equation of a Circle

The equation of a circle in its most generalized form is quite versatile and leads us to identify specific geometric properties. A circle's equation is often expressed in the form \(x^2 + y^2 + Dx + Ey + F = 0\). However, this form is not ideal for identifying the circle's center and radius.

By manipulating this expression, particularly through completing the square, we can rewrite it in a more insightful form. The equation can be rearranged to resemble

• \((x-h)^2 + (y-k)^2 = r^2\)

where \((h, k)\) is the center of the circle, and \(r\) is the radius. This format directly shows the circle's center and radius.

The process of reaching this concise form from the broader equation involves algebraic rearrangement and is crucial in enabling the direct extraction of geometric interpretations from the equation.

Standard Form of a Circle

The standard form of a circle's equation is a key point of reference when graphing or solving problems related to circles. This form is given by

• \((x-h)^2 + (y-k)^2 = r^2\)

where \((h, k)\) represents the center and \(r\) the radius of the circle. It becomes pivotal in understanding and utilizing the geometric features of a circle for further calculations or graphing purposes.

For example, to convert from a general form like \(x^2 + 2x + y^2 = 4\) to the standard form, completing the square reveals the transformed state

• \((x+1)^2 + y^2 = 3\)

This indicates a circle centered at \((-1, 0)\) with a radius of \(\sqrt{3}\).

The clarity provided by the standard form is apparent; it not only highlights the circle's position on the coordinate plane but also offers an immediate reference for calculating areas, tackling intersection points, or exploring tangents.