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 4. B. The slope of the tangent line is . C. The slope of the tangent line is -4. D. The slope of the tangent line is .

Answer 1

`\textsf{The\;slope\;of\;the\;tangent\;line\;is\;\;$-(1)/(4)$.}`

Explanation:

The slope-intercept form of a linear equation is given by

`\large\boxed{y=mx+b}`

where m is the slope, and b is the y-intercept.

Given that the equation of the diameter is y = 4x + 2, then its slope is m = 4.

The tangent line to a circle is always perpendicular to its radius, which means that the tangent line is also perpendicular to the diameter of the circle.

When two lines are perpendicular to each other, the product of their slopes is -1. Therefore:

`\begin{aligned}m{_\sf tangent} * m_(\sf diameter)& =-1\\\\m_(\sf tangent) * 4& =-1\\\\m_(\sf tangent)& = -(1)/(4)\end{aligned}`

Therefore, the slope of the tangent line is:

`\Large\boxed{\boxed{\sf Slope\; of\; the\; tangent\; line=-(1)/(4)}}`