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When circle [tex]\( P \)[/tex] is plotted on a coordinate plane, the equation of the diameter that passes through point [tex]\( Q \)[/tex] on the circle is [tex]\( y = 4x + 2 \)[/tex]. Which statement describes the equation of a line that is tangent to circle [tex]\( P \)[/tex] at point [tex]\( Q \)[/tex]?

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].



Answer :

GinnyAnswer
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].

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