91Ó°ÊÓ

A circle is a set of points in a plane that are all equidistant from a given point, known as the center. The standard equation for a circle with a center at

• (h, k) is \( (x-h)^2 + (y-k)^2 = r^2 \),
• where \( r \) is the radius of the circle.

For example, the equation \( x^2 + y^2 = 4 \) represents a circle centered at the origin (0,0) with radius 2, as the equation can be rewritten using:

• \( h = 0, k = 0 \) (the center),
• \( r^2 = 4 \) (indicating a radius \( r = 2 \)).

To solve problems involving circles, recognizing the equation of the circle and correctly identifying the center and radius is crucial. The center tells us the location of the circle on the coordinate plane, and the radius defines the size.

The perpendicular distance formula is a tool used to find the distance from a point to a line. In the context of circles, this can be especially helpful to confirm if a line is tangent to a circle. The formula is given as:
\[d = \frac{|Ax + By + C|}{\sqrt{A^2 + B^2}}\]Here,

• \( (x, y) \) is the point usually the center of the circle,
• \( Ax + By + C = 0 \) is the line equation.

For a line to be tangent to a circle, the perpendicular distance from the center of the circle to the line must equal the radius. In our exercise, the line we tested was:

• \( y = mx + 4 \)

Transforming it gives:

• \( -mx + y - 4 = 0 \)

By using

• \( A = -m \), \( B = 1 \), \( C = -4 \)

in the formula, the distance equation becomes \( \frac{4}{\sqrt{m^2 + 1}} = 2 \), including
the known radius \( r = 2 \).This distance formula is fundamental in assuring the line is precisely tangent at exactly one point.

The slope is a measure of the steepness of a line, represented by \( m \) in the line's equation, \( y = mx + b \). The slope is defined as the ratio of the change in the y-coordinate to the change in the x-coordinate between two points on the line:
\[ m = \frac{\Delta y}{\Delta x} \]Slopes of tangent lines to circles have unique attributes. If the line is tangent, its slope precisely matches one possible slope achieved when drawing a line through the given point and having just one point of intersection with the circle. In our exercise, we calculated slopes that are tangent using:

• \( m = \sqrt{3} \)
• \( m = -\sqrt{3} \)

This produces two different lines that only intersect the circle at one point each. The slope provides the inclination or direction, determining how the line rises or falls as it moves along the x-axis.