Question

question 20 of 25
this circle is centered at the origin, and the length of its radius is 5. what is
the equation of the circle?
a. $x^{2}+y^{2}=5$
b. $(x + 5)^{2}+(y - 5)^{2}=25$
c. $\frac{x^{2}}{5}+\frac{y^{2}}{5}=1$
d. $x^{2}+y^{2}=5^{2}$

Explanation:

Step1: Recall the standard equation of a circle

The standard equation of a circle with center \((h,k)\) and radius \(r\) is \((x - h)^{2}+(y - k)^{2}=r^{2}\).

Step2: Identify the center and radius

Since the circle is centered at the origin \((0,0)\) (so \(h = 0,k = 0\)) and radius \(r = 5\).

Step3: Substitute into the standard equation

Substitute \(h = 0,k = 0,r = 5\) into \((x - h)^{2}+(y - k)^{2}=r^{2}\), we get \((x-0)^{2}+(y - 0)^{2}=5^{2}\), which simplifies to \(x^{2}+y^{2}=5^{2}\).

For option A: \(x^{2}+y^{2}=5\) has radius \(\sqrt{5}
eq5\). For option B: \((x + 5)^{2}+(y - 5)^{2}=25\) has center \((-5,5)
eq(0,0)\). For option C: \(\frac{x^{2}}{5}+\frac{y^{2}}{5}=1\) is not the equation of a circle (the standard circle equation is \((x - h)^{2}+(y - k)^{2}=r^{2}\), not in the form of a sum of fractions equal to 1).

Answer:

D. \(x^{2}+y^{2}=5^{2}\)