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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 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]$\frac{1}{4}$[/tex].
B. The slope of the tangent line is -4.
C. The slope of the tangent line is 4.
D. The slope of the tangent line is [tex]$-\frac{1}{4}$[/tex].


Sagot :

To find the equation of a line that is tangent to circle [tex]\( P \)[/tex] at point [tex]\( Q \)[/tex], follow these steps:

1. Identify the slope of the diameter:
The equation of the diameter passing through point [tex]\( Q \)[/tex] is given by [tex]\( y = 4x + 2 \)[/tex]. This means the slope of the diameter is 4.

2. Determine the slope of the tangent line:
The tangent to a circle at any point is perpendicular to the radius (diameter) at that point. If two lines are perpendicular, the product of their slopes is -1. Therefore, if the slope of the diameter is 4, the slope of the tangent line, [tex]\( m_t \)[/tex], can be found using:
[tex]\[ 4 \cdot m_t = -1 \][/tex]
Solving for [tex]\( m_t \)[/tex]:
[tex]\[ m_t = -\frac{1}{4} \][/tex]

3. Compare with the given choices:
- A. The slope of the tangent line is [tex]\(\frac{1}{4}\)[/tex]. This is incorrect as our result shows the slope should be [tex]\(-\frac{1}{4}\)[/tex].
- B. The slope of the tangent line is -4. This is also incorrect because -4 is not the negative reciprocal of 4.
- C. The slope of the tangent line is 4. This is incorrect as 4 is the slope of the diameter, not the tangent.
- D. The slope of the tangent line is [tex]\(-\frac{1}{4}\)[/tex]. This is correct according to our calculation.

Therefore, the correct answer is:

D. The slope of the tangent line is [tex]\(-\frac{1}{4}\)[/tex].