Find an equation for a circle with center \( (-3,8) \), tangent (touching at one point) to the \( y \)-axis.

To find the equation of the circle, we need to determine its radius. The center of the circle is \((-3, 8)\). Since the circle is tangent to the \( y \)-axis, the distance from the center to the \( y \)-axis equals the radius. The distance from the center \((-3,8)\) to the \( y \)-axis (where \( x = 0 \)) is \( 3 \) units. Thus, the radius \( r \) is \( 3 \). The standard form of the equation of a circle is given by: \[ (x - h)^2 + (y - k)^2 = r^2 \] Where \((h, k)\) is the center. Plugging in the center and the radius, we have: \[ (x + 3)^2 + (y - 8)^2 = 3^2 \] Simplifying further, the equation of the circle is: \[ (x + 3)^2 + (y - 8)^2 = 9 \] Now, you've got a circle that’s just a hop away from the \( y \)-axis! But did you know? Circles are fascinating! They are not just found in geometry and math classes; their mathematical principles help in various fields like engineering, physics, and even in design. Circles create balance and harmony, which is why they often appear in logos and art! If you're curious about the applications of circles, consider how they play a role in wheel designs. The round shape allows for smooth rolling motion, showing the brilliance of circular geometry in everyday objects. From bicycles to automobiles, circles ensure efficiency and functionality in transportation!