To determine which statement describes the equation of a line that is tangent to circle [tex]\( P \)[/tex] at point [tex]\( Q \)[/tex], follow these steps:
1. Determine the slope of the diameter line:
- The equation of the diameter passing through point [tex]\( Q \)[/tex] is given by [tex]\( y = 4x + 2 \)[/tex].
- From this linear equation, you can directly observe that the slope of this line (denoted as [tex]\( m_d \)[/tex]) is [tex]\( 4 \)[/tex].
2. Find the slope of the tangent line:
- The slope of a tangent line at any point on a circle is perpendicular to the radius (or diameter) at that point.
- To find the slope of a line perpendicular to another line, you take the negative reciprocal of the original slope.
- Therefore, the slope of the tangent line (denoted as [tex]\( m_t \)[/tex]) that is perpendicular to a slope of [tex]\( 4 \)[/tex] would be:
[tex]\[
m_t = -\frac{1}{4}
\][/tex]
3. Verify the correct statement:
- The question provides four options for the slope of the tangent line:
- A. The slope of the tangent line is [tex]\( -4 \)[/tex].
- B. The slope of the tangent line is [tex]\(-\frac{1}{4} \)[/tex].
- C. The slope of the tangent line is [tex]\(\frac{1}{4} \)[/tex].
- D. The slope of the tangent line is [tex]\( 4 \)[/tex].
Taking the calculated slope [tex]\( -\frac{1}{4} \)[/tex], the correct option is:
The answer is B. The slope of the tangent line is [tex]\( -\frac{1}{4} \)[/tex].
