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Chapter 0: Problem 30

If the line passing through the points \((-2,4)\) and \((1, a)\) is perpendicular to the line passing through the points \((a+4,8)\) and \((3,-4)\), what must the value of \(a\) be?

Short Answer

Expert verified
The value of \(a\) is -5.

Step by step solution

01

Find the slope of the line passing through (-2, 4) and (1, a)

To find the slope, use the slope formula: \[m = \frac{y2 - y1}{x2 - x1}\] Plugging in the points, we get: \[m_1 = \frac{a - 4}{1 - (-2)}\]
02

Find the slope of the line passing through (a + 4, 8) and (3, -4)

Apply the same slope formula for these points as well: \[m_2 = \frac{-4 - 8}{3 - (a+4)}\]
03

Set the product of the slopes to -1

Given that the lines are perpendicular, the product of their slopes will be -1: \[m_1 \cdot m_2 = -1\]
04

Substitute the slopes and solve for a

Now, plug in the slopes we found earlier and solve for \(a\): \[\frac{a - 4}{1 - (-2)} \cdot \frac{-4 - 8}{3 - (a+4)} = -1\] Simplify and solve for \(a\): \[(a - 4)(-12) = 3(1 - a)\] Expanding the equation, we get: \[-12a + 48= 3 - 3a\] Combining the terms, we get: \[9a = -45\] Now divide by 9: \[a = -5\] So the value of \(a\) is -5.

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