Question 184086
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First thing to do is find the coordinates of the mid-point of line segment *[tex \Large \overline{AB}].  That is because the perpendicular bisector of the line segment intersects the line segment at the mid-point by definition of a perpendicular bisector.


Use the mid-point formula:


*[tex \LARGE \text{          }\math (x_m,y_m) = \left(\frac{x_1 - x_2}{2},\frac{y_1 - y_2}{2}\right)]


where *[tex \Large (x_1,y_1)] and *[tex \Large (x_2,y_2)] are the endpoints of the line segment.


Next you need the slope, <i>m</i>, of the line that contains the line segment:


*[tex \LARGE \text{          }\math m = \frac{y_1 - y_2}{x_1 - x_2}]


where *[tex \Large (x_1,y_1)] and *[tex \Large (x_2,y_2)] are the endpoints of the line segment.


The slope of a perpendicular is the negative reciprocal of the slope of the given line, that is:


*[tex \LARGE \text{          }\math L_1 \perp L_2 \ \ \Leftrightarrow\ \ m_1 = -\frac{1}{m_2}]


So, calculate *[tex \Large -\frac{1}{m}] using the value of the slope, <i>m</i>, determined above.


Then use this new slope value plus the fact that the calculated midpoint, *[tex \Large (x_m,y_m)] is a point on the desired perpendicular to provide the given information for the point-slope form of the equation of a straight line, namely:


*[tex \LARGE \text{          }\math y - y_m = -\frac{1}{m}(x - x_m)]


Make the appropriate substitutions for *[tex \Large (x_m,y_m)] and <i>m</i> and then solve the equation for <i>y</i> in terms of <i>x</i> to obtain the desired slope-intercept form, *[tex \Large y = m_p x + b]


The resulting *[tex \Large m_p] and <i>b</i> will be the answers requested.



John
*[tex \LARGE e^{i\pi} + 1 = 0]
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