Find the value of p, if the line which passes through (-1, -p) and (-2p, 2) is p...

In this problem, we're tasked with finding the value of \(p\) such that the line passing through the points (-1, -p) and (-2p, 2) is parallel to the line defined by the equation 2y + 8x - 17 = 0. First, let's understand what it means for two lines to be parallel in the coordinate plane. Two lines are parallel if and only if their slopes are equal. The slope of a line is given by the change in y-coordinates divided by the change in x-coordinates between any two points on the line. This is often expressed as \(m = \frac{y_2 - y_1}{x_2 - x_1}\).

For the line 2y + 8x - 17 = 0, we can rearrange it into the form \(y = mx + c\) to find its slope. This gives us \(y = -4x + \frac{17}{2}\). So, the slope of this line is -4.

Next, we can find the slope of the line passing through the points (-1, -p) and (-2p, 2). Using the formula for slope, we get \(m = \frac{2 - (-p)}{-2p - (-1)} = \frac{2 + p}{1 - 2p}\).

Now, for the lines to be parallel, these two slopes must be equal. That gives us the equation \(-4 = \frac{2 + p}{1 - 2p}\). Solving this equation for \(p\) will give us the desired value.

If we cross-multiply, we get \(-4 + 8p = 2 + p\). Simplifying this gives us \(7p = 6\), or \(p = \frac{6}{7}\).

So the correct answer is \(p = \frac{6}{7}\), which corresponds to Option A.