Find the coordinates of the foot of the perpendicular from the point (1, 2) to t

Find the coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x - 3y + 6 = 0.

Find the coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x - 3y + 6 = 0.

Step 1: Identify the point from which we want to drop the perpendicular. This point is (1, 2).
Step 2: Write down the equation of the line to which we are dropping the perpendicular. The line is given as 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 (1, 2). The equation is y - 2 = -3/2(x - 1).
Step 7: Simplify the equation of the perpendicular line. This gives us y = -3/2x + 3.5.
Step 8: Now, we need to find the intersection of the original line and the perpendicular line. Set the two equations equal to each other: (2/3)x + 2 = -3/2x + 3.5.
Step 9: Solve for x. This involves combining like terms and isolating x.
Step 10: Once you find x, substitute it back into either line equation to find the corresponding y-coordinate.
Step 11: The coordinates you find will be the foot of the perpendicular from the point (1, 2) to the line.

Foot of Perpendicular – Finding the point on a line that is closest to a given point, which forms a right angle with the line.
Line Equation – Understanding how to manipulate and interpret the equation of a line in standard form.
Distance from a Point to a Line – Using geometric principles to calculate the shortest distance from a point to a line.