Problem 33 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the circle. Identify its center and radius. $$x^{2}+2 x+y^{2}-35=0$$

Step-by-Step Solution

Answer

The circle with equation \(x^{2}+2x+y^{2}-35=0\) has its center at (-1, 0) and a radius of 6 units.

Step 1 Title: Understanding the Circle Equation

Understand that the given equation \(x^{2}+2 x+y^{2}-35=0\) is a circle equation, but it is not in the standard form. The standard form of a circle's equation is \( (x - h)^2 + (y - k)^2 = r^2 \), where (h, k) is the center and r is the radius.

Step 2 Title: Expressing the equation in standard form

To express the given equation in the standard form, complete the square for x. Indeed, our equation can be rewritten as \((x + 1)^2 + y^2 - 36 = 0\). Then, it can be rearranged to obtain the equation in standard form: \((x + 1)^2 + y^2 = 36\).

Step 3 Title: Identifying center and radius

Now by comparing our equation with the standard form, we can identify the center and radius of the circle. The center of the circle is (-1, 0) and the radius is \(\sqrt{36}\), which is 6.

Step 4 Title: Drawing the circle

Now, you can draw a circle with center at (-1, 0) and radius of 6 units on the x-y plane. Mark the center with a point and use a compass or a circular tool to draw a circle with the given radius. The circle crosses the x-axis at two points, (-7, 0) and (5, 0), and the y-axis at two points, (-1, -6) and (-1, 6).

Key Concepts

Completing the Square

Completing the square is a useful algebraic method to rewrite a quadratic expression so that it resembles a perfect square trinomial. Let's try to understand it step by step using our circle's equation.

The given circle equation is:

• \[ x^{2} + 2x + y^{2} - 35 = 0 \]

To complete the square for the term involving \(x\), we first focus on \(x^{2} + 2x\). The goal is to have this expression look like \((x+a)^2\).

• Take half of the coefficient of \( x \) (which is 2), then square it. This means: \( \left(\frac{2}{2}\right)^2 = 1 \).
• Add and subtract this number (1) within the equation to keep it balanced.

So,

• \(x^{2} + 2x + 1 - 1 = (x+1)^2 - 1\).

Then, replace \(x^{2} + 2x\) with \((x+1)^2 - 1\). Now the equation reflects a neat completed square:

• \((x + 1)^2 + y^2 - 36 = 0\)

Completing the square is very helpful when trying to rewrite the equation into the standard form of a circle equation. But, remember that even though it might seem abstract at first, practice makes it much simpler.

Standard Form of a Circle

A circle has a specific equation format known as the standard form. This makes it easier to identify key components like the center and radius of the circle.

The standard form for a circle's equation is:

• \[ (x - h)^2 + (y - k)^2 = r^2 \]

Here,

• \((h, k)\) represents the center of the circle.
• \(r\) is the radius.

To convert the original equation, \(x^{2}+2x+y^{2}-35=0\), into standard form, we follow these steps:

• Complete the square for \(x\), transforming the equation to \((x + 1)^2 + y^2 = 36\).
• This format helps us see the circle's characteristics more clearly.

The equation now resembles the standard format, allowing us to readily identify:

• The center of the circle at \((-1,0)\).
• The radius derived from \(r^2 = 36\).

Mastering the standard form allows anyone working with circles to easily interpret and manipulate them.

Radius of a Circle

The radius of a circle is simply the distance from the center to any point on the circle's edge. Knowing how to identify this helps in sketching and understanding circles.

From the standard form of a circle equation, \((x - h)^2 + (y - k)^2 = r^2\), the term \(r^2\) represents the square of the circle's radius.

• When we rewrote our circle equation as \((x + 1)^2 + y^2 = 36\), the 36 corresponds to \(r^2\).
• To find \(r\), you take the square root of 36.
• This calculation gives us \(r = \sqrt{36} = 6\).

Understanding the concept of radius is crucial as it helps in practical applications, such as drawing the circle. Knowing the radius allows you to determine how far the edge extends from the center, ensuring your circle is accurate. This fundamental element of circles is pivotal in diverse math problems and real-world tasks involving circular shapes.