Problem 8 - 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 (0, 0) and the radius is \(\sqrt{\frac{2}{3}}\).

Step 1: Rewrite the equation in the standard form

To rewrite the equation in the standard form, we first need to divide both sides of the equation by 15. The standard form is \((x-a)^{2}+(y-b)^{2}=r^2\), where the center of the circle is (a, b) and r is the radius. Dividing both sides by 15 gives: $$x^{2}+y^{2}=\frac{2}{3}$$ Now, the equation is in the standard form.

Step 2: Identify the center and radius

Since our equation is \(x^{2}+y^{2}=\frac{2}{3}\), we can see that there are no specific x or y terms. Therefore, the circle's center (a, b) is (0, 0). The radius, r, can be found by taking the square root of the constant term on the right side of the equation. In this case, r = \(\sqrt{\frac{2}{3}}\). So, the center of the circle is (0, 0) and the radius is \(\sqrt{\frac{2}{3}}\).