**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

The correct answer is A.

### 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**.

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2y + 8× - 17= 0

Find the gradient by making y the subject of formula. Divide all through by 2

8x /-2 -17/-2 = -2y/-2

y= -4x +17/2

So m= -4

Using the formula M= y2 - y1/ x2 - x1

M=-4, x1= 1, x2=2p, y1=-p, y2=2

Therefore substitute the values into the formula

-4= 2-(-p)/ -2p-(-1)

-4=2+p/-2p+1

Cross multiply

-4(-2p+1) = 2+p

8p-4 = 2+p

8p-p =2+4

7p = 6

Therefore p = 6/7