The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:

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Q: The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:

Solution: Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).

Steps: 13

Step 1: Identify the point from which we want to drop a perpendicular. This point is (3, 4).
Step 2: Write down the equation of the line, which is 2x + 3y - 6 = 0.
Step 3: Rearrange the line equation into slope-intercept form (y = mx + b) to find the slope. The line can be rewritten as 3y = -2x + 6, or y = -2/3 x + 2.
Step 4: Determine the slope of the line, which is -2/3. The slope of the perpendicular line will be the negative reciprocal, which is 3/2.
Step 5: Use the point-slope form of the line equation to write the equation of the perpendicular line that passes through (3, 4). The equation is y - 4 = (3/2)(x - 3).
Step 6: Simplify the equation of the perpendicular line to find its y-intercept form. This gives us y = (3/2)x - (9/2) + 4, or y = (3/2)x - (1/2).
Step 7: Now, we have two equations: the original line (2x + 3y - 6 = 0) and the perpendicular line (y = (3/2)x - (1/2)).
Step 8: Substitute the expression for y from the perpendicular line into the original line equation to find the x-coordinate of the foot of the perpendicular.
Step 9: Solve for x in the equation 2x + 3((3/2)x - (1/2)) - 6 = 0.
Step 10: After solving, you will find x = 2.
Step 11: Substitute x = 2 back into the equation of the perpendicular line to find the corresponding y-coordinate.
Step 12: After substituting, you will find y = 0.
Step 13: Therefore, the coordinates of the foot of the perpendicular are (2, 0).