To determine which statement describes the correct equation of a line that is tangent to circle P at point Q, let's think about the relationship between a circle and a tangent line at a point on the circle.
1. The tangent line to a circle at a given point is always perpendicular to the radius (or diameter) of the circle at that point.
2. The equation of the diameter given is [tex]\( y = 4x + 2 \)[/tex].
In this form, [tex]\( y = mx + b \)[/tex] (where [tex]\( m \)[/tex] is the slope), we can see that the slope ([tex]\( m \)[/tex]) of the diameter is 4.
3. Since the tangent line is perpendicular to the diameter at point Q, the slope of the tangent line will be the negative reciprocal of the slope of the diameter.
- The negative reciprocal of 4 is [tex]\( -\frac{1}{4} \)[/tex].
Therefore, the slope of the tangent line is [tex]\( -\frac{1}{4} \)[/tex].
Thus, the correct statement describing the equation of a line that is tangent to circle P at point Q is:
C. The slope of the tangent line is [tex]\( -\frac{1}{4} \)[/tex].
