When circle P is plotted on a coordinate plane, the equation of the diameter that passes through point Q on the circle is $y=4 x+2$. Which statement describes the equation of a line that is tangent to circle $P$ at point $Q$?
A. The slope of the tangent line is $\frac{1}{4}$.
B. The slope of the tangent line is $-\frac{1}{4}$.
C. The slope of the tangent line is 4
D. The slope of the tangent line is $- 4$
Answer (1)
The slope of the diameter is 4.
The tangent line is perpendicular to the diameter.
The slope of the tangent line is the negative reciprocal of the diameter's slope.
The slope of the tangent line is − 4 1 .
Explanation
Problem Analysis The problem states that a circle P is plotted on a coordinate plane, and the equation of the diameter passing through point Q on the circle is given by y = 4 x + 2 . We are asked to find the slope of the line tangent to the circle at point Q .
Find the slope of the diameter The equation of the diameter is in slope-intercept form, y = m x + b , where m is the slope and b is the y-intercept. In this case, the slope of the diameter is 4 .
Recognize the relationship between tangent and diameter The tangent line to a circle at a point is perpendicular to the radius (or diameter) at that point. Therefore, the tangent line at point Q is perpendicular to the diameter at point Q .
Calculate the slope of the tangent line The slopes of perpendicular lines are negative reciprocals of each other. If the slope of the diameter is m 1 , and the slope of the tangent line is m 2 , then m 1 ⋅ m 2 = − 1 . Since m 1 = 4 , we have 4 ⋅ m 2 = − 1 . Solving for m 2 , we get m 2 = − 4 1 .
State the final answer Therefore, the slope of the tangent line to circle P at point Q is − 4 1 .
Examples
Imagine you're designing a circular fountain in a park. The path of the water jet at any point on the circle needs to be tangent to the circle at that point. Knowing the equation of the diameter, you can easily calculate the slope of the tangent, ensuring the water jet flows in the desired direction. This principle ensures that the water feature is both aesthetically pleasing and geometrically sound.
