Final answer:
The equation of the line passing through the point that divides AB in the ratio of 3:2 and is perpendicular to AB is y = 3x - 5.8. None of the given options (A, B, C, D) match this equation.
Step-by-step explanation:
The student has been asked to find the equation of the line that divides the segment AB in the ratio of 3:2 and is perpendicular to it, where A(-3, 4) and B(3, 2). First, we need to find the coordinates of the point dividing AB in the given ratio using the section formula. For the ratio 3:2, let the dividing point be P(x,y). The coordinates of P are given by:
x = [(3)(-3) + (2)(3)] / (3 + 2) = (9 + 6) / 5 = 3
y = [(3)(4) + (2)(2)] / (3 + 2) = (12 + 4) / 5 = 16 / 5 = 3.2
So, P has coordinates (3, 3.2). The slope of the line through A and B can be calculated using the formula (y2 - y1) / (x2 - x1). For A and B, the slope (m) equals:
m = (2 - 4) / (3 - (-3)) = (-2) / 6 = -1/3. The slope of the line perpendicular to AB will be the negative reciprocal, 3. Now, using the point-slope form of the line equation y - y1 = m(x - x1), where m is the slope and (x1,y1) is the point on the line, the equation of the line is:
y - 3.2 = 3(x - 3)
Distributing and rearranging gives:
y = 3x - 9 + 3.2
y = 3x - 5.8
The line passing through the dividing point P and perpendicular to AB does not match any of the given options (A, B, C, D). Hence, none of the given options is correct.