Problem 64 - Contemporary Precalculus

Question

Find the center and radius of the circle whose equation is given. $$15 x^{2}+15 y^{2}=10$$

Step-by-Step Solution

Answer

Answer: The center of the circle is at the origin (0, 0) and its radius is $r = \sqrt{\frac{2}{3}}$.

Step 1: Divide the equation by 15

To make the coefficients of \(x^2\) and \(y^2\) equal to 1, divide the entire equation by 15: $$\frac{15x^2}{15} + \frac{15y^2}{15} = \frac{10}{15}$$ This simplifies to: $$x^2 + y^2 = \frac{2}{3}$$

Step 2: Identify the circle's center and radius

Now that the equation is in the form $$x^2+y^2=r^2$$, we can compare it to the standard form of a circle equation $$(x-h)^2+(y-k)^2=r^2$$ In this case, since there are no \(x\) or \(y\) terms in the equation (like \((x-h)\) or \((y-k)\)), it means that the center of the circle is at the origin, which means h = 0 and k = 0. The circle's center is (h, k) = (0, 0). The right side of the equation is the radius squared, so $$r^2 = \frac{2}{3}$$. To find the radius, we take the square root of both sides: $$r = \sqrt{\frac{2}{3}}$$

Step 3: Write the final answer

The center of the circle is at the origin (0, 0) and its radius is $$r = \sqrt{\frac{2}{3}}$$.

Key Concepts

Standard Form of a Circle

When dealing with circles in mathematics, one of the most useful equations is the Standard Form of a Circle. It's typically written as \[(x-h)^2 + (y-k)^2 = r^2,\] where:

• g represents the x-coordinate of the circle's center.
• k indicates the y-coordinate of the center.
• r stands for the radius of the circle.

This particular format is incredibly useful because it directly gives us information about where the circle is located on the coordinate plane and how large it is.

If you're given a circle equation in another form, like the one in the original problem \(x^2 + y^2 = \frac{2}{3},\) a quick transformation can put it into standard form, which is what we did after dividing by 15. This simplified 'centeress' form is actually a special case of the circle equation where the circle is centered at the origin.

Center of a Circle

The center of a circle offers a vital reference point, allowing you to understand the circle's exact position on the plane. For any equation given in the standard form \((x-h)^2 + (y-k)^2 = r^2,\)the coordinates \((h, k)\)are always the circle's center. This equation format tells us that the horizontal distance from the center is h and the vertical distance is k.

If the equation doesn't feature any terms for x or y, like \(x^2 + y^2 = \frac{2}{3},\) it means the circle's center is simply at the origin: \((0, 0)\).In such cases, both h and k equal zero. Recognizing this makes it easier to visualize and solve problems involving circles, and helps set a foundation for graphing them as well.

Radius of a Circle

The radius is a crucial aspect of any circle—it's the distance from the center point to any edge on the circle.

Mathematically, in the equation \((x-h)^2 + (y-k)^2 = r^2,\)it's denoted as r and represents the square root of the right-hand side, \(r^2.\)This means whenever you have a value as \(r^2,\)simply take the square root to determine the radius itself.

For example, in \(x^2 + y^2 = \frac{2}{3},\)the right side \(= \frac{2}{3}\), so the radius, called \(r,\)becomes \(\sqrt{\frac{2}{3}}.\)This calculation makes it simple to draw and understand the size of the circle, no matter what center it may have. Being comfortable finding the radius is key to solving more complex circle-related problems.