...sub-question (c): Center at \((16, 9)\) and radius \(\frac{3}{4}\)...
...step subtitle: Write the standard equation of a circle...
The standard equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
...step subtitle: Substitute the given center and radius...
Substitute \(h = 16\), \(k = 9\), and \(r = \frac{3}{4}\) into the equation:
\[
(x - 16)^2 + (y - 9)^2 = \left(\frac{3}{4}\right)^2
\]
...step subtitle: Simplify the equation...
Simplify \(\left(\frac{3}{4}\right)^2\) to \(\frac{9}{16}\):
\[
(x - 16)^2 + (y - 9)^2 = \frac{9}{16}
\]
...sub-answer: The equation of the circle is \(\boxed{(x - 16)^2 + (y - 9)^2 = \frac{9}{16}}\).
...sub-question (d): Center at \((-1, 2)\) and radius \(\sqrt{3}\)...
...step subtitle: Write the standard equation of a circle...
The standard equation of a circle with center \((h, k)\) and radius \(r\) is:
\[
(x - h)^2 + (y - k)^2 = r^2
\]
...step subtitle: Substitute the given center and radius...
Substitute \(h = -1\), \(k = 2\), and \(r = \sqrt{3}\) into the equation:
\[
(x - (-1))^2 + (y - 2)^2 = (\sqrt{3})^2
\]
...step subtitle: Simplify the equation...
Simplify \((x - (-1))^2\) to \((x + 1)^2\) and \((\sqrt{3})^2\) to \(3\):
\[
(x + 1)^2 + (y - 2)^2 = 3
\]
...sub-answer: The equation of the circle is \(\boxed{(x + 1)^2 + (y - 2)^2 = 3}\).
...summary of all sub-answers here...
The equation of the circle for sub-question (c) is \(\boxed{(x - 16)^2 + (y - 9)^2 = \frac{9}{16}}\).
The equation of the circle for sub-question (d) is \(\boxed{(x + 1)^2 + (y - 2)^2 = 3}\).
Answered
