Problem 18 - Algebra and Trigonometry Real Mathematics, Real People

Question

Identify the center and radius of the circle. $$(x+7)^{2}+(y-3)^{2}=32$$

Step-by-Step Solution

Answer

The center of the circle is (-7, 3) and the radius of the circle is approximately 5.66.

Step 1: Identify the Center

The center of the circle is given by \((h, k)\), where \(h\) and \(k\) are the opposite of the values inside the brackets of the standard equation. So, the center of the circle is \((-7, 3)\)

Step 2: Identify the Radius

The radius of the circle is given by \(r\), which is the square root of the value on the right side of the equation. The square root of 32 gives you approximately 5.66. So, the radius of the circle is \(5.66\).

Key Concepts

Standard Form of a Circle

Understanding the standard form equation of a circle is vital for dissecting all the important characteristics of a circle with ease. The standard form for the equation of a circle is given by the formula \( (x-h)^2 + (y-k)^2 = r^2 \), where \( (h, k) \) represents the center of the circle, and \( r \) represents the radius.

If you've ever wondered why it looks like this, it's essentially an algebraic version of the Pythagorean theorem applied to any point \( (x, y) \) on the circle's circumference, as measured from its center. In the given example, \( (x+7)^2 + (y-3)^2 = 32 \), a quick reconfiguration can help us notice that it aligns perfectly with the standard form, immediately hinting at the position of the center and the square of the radius.

A solid grasp of this form allows you to sketch a circle provided its equation or formulate the equation given specific points. When faced with more complex equations, rearranging terms to achieve this standard form will often simplify your geometrical endeavours.

Identifying the Center of a Circle

Pinpointing the center is a key skill when analyzing circle equations. It's a straightforward process once the equation is in standard form. In the equation \( (x-h)^2 + (y-k)^2 = r^2 \), the center is hiding in plain sight—just take the opposite signs of \( h \) and \( k \).

Remember, coordinate signs in maths are like directions; taking the 'opposite' means changing the direction. For instance, the center from our example, \( (x+7)^2 + (y-3)^2 = 32 \), is at \( (-7, 3) \) because we switch the signs inside the parentheses.

Why does this work? It comes down to the circle's symmetry. To shift a circle's center to a point \( (h, k) \) away from the origin \( (0, 0) \) coordinates become \( (x-h) \) and \( (y-k) \)—hence, the 'opposite' signs when you locate the center. It's a neat little trick to always have up your sleeve!

Calculating the Radius of a Circle

The radius is the heart of the circle—everything revolves around it, quite literally! Calculating the radius when you have a circle's equation in standard form is as simple as finding the square root of the right side of the equation. So, if our equation is \( (x+7)^2 + (y-3)^2 = 32 \) and the standard form is \( (x-h)^2 + (y-k)^2 = r^2 \) by comparison, \( r^2 = 32 \).

To unveil the radius, we simply calculate the square root of 32, which is about 5.66. This is the measure of the distance from the center \( (-7, 3) \) to any point on the circle's edge.

Always keep in mind the power of this formula relation—it's not only useful for plotting or drawing circles but also a cornerstone for solving problems involving tangents, chords, and even calculus problems that deal with circles. The radius unlocks the circle's secrets, so knowing how to find it efficiently can serve you well across many areas of mathematics.