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

### Question

Find the value of p, if the line which passes through (-1, -p) and (-2p, 2) is parallel to the line 2y + 8x - 17 = 0.

### Options

A)
6/7 B)
4/7
C)
2/5
D)
-6/7

### Explanation:

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.