## Answer :

Here's a detailed, step-by-step approach to solving the problem:

1.

**Identify the slope of the given diameter**:

The equation of the diameter that passes through point [tex]\(Q\)[/tex] is given as [tex]\(y = 4x + 2\)[/tex]. This equation is in the slope-intercept form [tex]\(y = mx + b\)[/tex], where [tex]\(m\)[/tex] is the slope of the line. From the equation, we see that the slope [tex]\(m\)[/tex] of the diameter is [tex]\(4\)[/tex].

2.

**Understand the geometric relationship**:

In coordinate geometry, the tangent to a circle at any point is perpendicular to the radius (or a diameter) of the circle at that point. Therefore, the slope of the tangent line at point [tex]\(Q\)[/tex] will be the negative reciprocal of the slope of the diameter.

3.

**Calculate the negative reciprocal of the slope**:

The slope of the diameter is [tex]\(4\)[/tex]. The negative reciprocal of [tex]\(4\)[/tex] is calculated as follows:

[tex]\[ \text{Negative Reciprocal} = -\frac{1}{4} \][/tex]

4.

**Determine which statement matches the calculated slope**:

The slope of the tangent line at point [tex]\(Q\)[/tex] is [tex]\(-\frac{1}{4}\)[/tex]. This corresponds to option [tex]\(B\)[/tex].

Therefore, the correct answer is:

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