Final answer:
The equation of the line perpendicular to the given line and passing through the point (2,-6) is y = 7/6x - 25/3.
Step-by-step explanation:
To find the equation of the line that is perpendicular to another, we first need to determine the slope of the given line. The slope (m) of the line passing through the points (-3,4) and (4,-2) is calculated using the formula m = (Y2 - Y1) / (X2 - X1). Plugging in the coordinates, we get m = (-2 - 4) / (4 - (-3)) = -6 / 7.
Now since the required line is perpendicular to this line, its slope will be the negative reciprocal of -6/7, which is 7/6. Using the point-slope form of the equation y - y1 = m(x - x1), and the point (2, -6), we get y - (-6) = 7/6(x - 2). Simplifying this, we get the equation of the perpendicular line as y + 6 = 7/6x - 7/3. Taking '6' to the other side, we get y = 7/6x - 25/3, which is the desired equation of the line.