To find the equation of a line that is perpendicular to the line passing through the points A(-8,0) and B(0,4) and shares the same y-intercept, we can follow these steps:
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Find the slope of the line AB: The formula for the slope \(m\) between two points \((x_1, y_1)\) and \((x_2, y_2)\) is given by: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting points A(-8,0) and B(0,4): \[ m = \frac{4 - 0}{0 - (-8)} = \frac{4}{8} = \frac{1}{2} \]
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Find the slope of the perpendicular line: The slope of a line that is perpendicular to another line is the negative reciprocal of the original line's slope. Since the slope of line AB is \(\frac{1}{2}\), the slope of the perpendicular line will be: \[ m_{\perpendicular} = -\frac{1}{\left(\frac{1}{2}\right)} = -2 \]
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Determine the y-intercept: The y-intercept of the original line (line AB) can be found using the point B(0,4). Since the line shares the same y-intercept, the y-intercept of the perpendicular line is also 4, which means the point is (0,4).
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Write the equation of the perpendicular line: Now we use the point-slope form of a line, which is given by: \[ y - y_1 = m(x - x_1) \] Using the point (0, 4) and the slope -2: \[ y - 4 = -2(x - 0) \] Simplifying this: \[ y - 4 = -2x \] \[ y = -2x + 4 \]
Thus, the equation of the line that is perpendicular to the line through points A and B and shares the same y-intercept is: \[ y = -2x + 4 \]
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