Find the coordinates of the foot of the perpendicular from the point (3, 4) to t

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

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

Correct Answer: (3, 2)

Step 1: Identify the point from which we want to drop the 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 equation becomes 3y = -2x + 6, or y = -2/3 x + 2.
Step 4: Determine the slope of the line. The slope (m) is -2/3.
Step 5: Find the slope of the perpendicular line. The slope of the perpendicular line is the negative reciprocal of -2/3, which is 3/2.
Step 6: 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 7: Simplify the equation of the perpendicular line. This gives us y = (3/2)x - (9/2) + 4, or y = (3/2)x - (1/2).
Step 8: Now, we have two equations: the original line (2x + 3y - 6 = 0) and the perpendicular line (y = (3/2)x - (1/2)).
Step 9: 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 10: Solve for x in the equation 2x + 3((3/2)x - (1/2)) - 6 = 0.
Step 11: After solving, you will find x = 3.
Step 12: Substitute x = 3 back into the equation of the perpendicular line to find the y-coordinate.
Step 13: This gives us y = (3/2)(3) - (1/2) = 4.5 - 0.5 = 4.
Step 14: The coordinates of the foot of the perpendicular are (3, 2).

Foot of Perpendicular – Finding the point on a line that is closest to a given point, which involves using the formula for the foot of the perpendicular from a point to a line.
Line Equation – Understanding how to manipulate and interpret the equation of a line in standard form.
Distance from a Point to a Line – Applying geometric principles to calculate the shortest distance from a point to a line.