Question

question 1 of 25
which of the following equations correctly represents a circle centered at the
origin with a radius of 7?

a. ( x^2 + y^2 = 7^2 )

b. ( (x - 7)^2 + (y - 7)^2 = 7^2 )

c. ( (x - 7)^2 + y^2 = 49 )

d. ( x^2 + y^2 = 7 )

Explanation:

Step1: Recall the circle equation formula

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

The circle is centered at the origin, so \(h = 0\) and \(k = 0\), and the radius \(r = 7\).

Step3: Substitute into the formula

Substituting \(h = 0\), \(k = 0\), and \(r = 7\) into the standard equation, we get \((x - 0)^2+(y - 0)^2=7^2\), which simplifies to \(x^2 + y^2=7^2\) (since \(7^2 = 49\), but \(7^2\) is also correct in form). Now we check each option:

• Option A: \(x^2 + y^2=7^2\) matches our derived equation.
• Option B: \((x - 7)^2+(y - 7)^2=7^2\) has center \((7,7)\), not the origin, so it's incorrect.
• Option C: \((x - 7)^2+y^2 = 49\) has center \((7,0)\), not the origin, so it's incorrect.
• Option D: \(x^2 + y^2=7\) has radius \(\sqrt{7}\), not 7, so it's incorrect.

Answer:

A. \(x^2 + y^2=7^2\)