To determine the correct equation of a circle with a radius [tex]\( r \)[/tex] and center at [tex]\( (h, v) \)[/tex], we need to recall the general form of the equation of a circle in a Cartesian plane.
A circle centered at [tex]\( (h, v) \)[/tex] with radius [tex]\( r \)[/tex] can be described by the equation:
[tex]\[ (x - h)^2 + (y - v)^2 = r^2 \][/tex]
This equation states that any point [tex]\( (x, y) \)[/tex] on the circle is a distance [tex]\( r \)[/tex] away from the center [tex]\( (h, v) \)[/tex]. Let's analyze each given option:
A. [tex]\((x - h)^2 + (y - v)^2 = r^2\)[/tex]
This equation matches the form we identified for the equation of a circle, so it is correct.
B. [tex]\((x + h)^2 + (y + y)^2 = r^2\)[/tex]
This equation is flawed because it incorrectly uses [tex]\( +h \)[/tex] instead of [tex]\( -h \)[/tex] and seems to have a typographical error [tex]\( y + y \)[/tex] instead of [tex]\( y - v \)[/tex].
C. [tex]\((x - v)^2 + (y - h)^2 = r^2\)[/tex]
This equation incorrectly swaps the [tex]\( h \)[/tex] and [tex]\( v \)[/tex] variables, which misplaces the center coordinates. This cannot be the correct equation of our circle.
D. [tex]\(h^2 + v^2 = r^2\)[/tex]
This is not even in the form of an equation of a circle involving [tex]\( x \)[/tex] and [tex]\( y \)[/tex]. It is irrelevant to the problem at hand.
Thus, the correct equation for a circle with radius [tex]\( r \)[/tex] and center at [tex]\( (h, v) \)[/tex] is:
Option A: [tex]\((x - h)^2 + (y - v)^2 = r^2\)[/tex]
