To determine the slope of the tangent line to circle [tex]\( P \)[/tex] at point [tex]\( Q \)[/tex], given that the equation of the diameter that passes through point [tex]\( Q \)[/tex] on the circle is [tex]\( y = 4x + 2 \)[/tex]:
1. Identify the slope of the diameter:
The equation of the diameter is given in the slope-intercept form [tex]\( y = mx + b \)[/tex], where [tex]\( m \)[/tex] represents the slope. In the equation [tex]\( y = 4x + 2 \)[/tex], the slope ([tex]\( m \)[/tex]) is 4.
2. Determine the slope of the tangent line:
The slope of the tangent line to a circle at a given point is the negative reciprocal of the slope of the diameter passing through that point. Therefore, the slope of the tangent line is the negative reciprocal of 4.
[tex]\[
\text{slope of tangent line} = -\frac{1}{4}
\][/tex]
3. Select the correct statement:
We found that the slope of the tangent line is [tex]\(-\frac{1}{4}\)[/tex].
Therefore, the correct statement is:
B. The slope of the tangent line is [tex]\(-\frac{1}{4}\)[/tex].
