Question

when circle p is plotted on a coordinate plane, the equation of the diameter that passes through point q on the circle is y = 4x + 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 $-4$.

Explanation:

Step1: Recall the property of tangent to a circle

The tangent to a circle at a point is perpendicular to the radius (or diameter) at that point. If two lines are perpendicular, the product of their slopes is -1. Let the slope of the diameter (given line) be \(m_1\) and the slope of the tangent line be \(m_2\), then \(m_1\times m_2=- 1\).

Step2: Find the slope of the given diameter line

The equation of the diameter is \(y = 4x+2\), which is in the slope - intercept form \(y=mx + c\) (where \(m\) is the slope and \(c\) is the y - intercept). So, the slope of the diameter line \(m_1 = 4\).

Step3: Calculate the slope of the tangent line

Using the formula for perpendicular lines \(m_1\times m_2=-1\). Substitute \(m_1 = 4\) into the formula:
\(4\times m_2=-1\)
Solve for \(m_2\): \(m_2=-\frac{1}{4}\)

Answer:

A. The slope of the tangent line is \(-\frac{1}{4}\)