The coordinates of the foot of the perpendicular from the point (1, 2) to the li

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Question: The coordinates of the foot of the perpendicular from the point (1, 2) to the line 3x + 4y = 12 are:

Options:

Correct Answer: (2, 1)

Solution:

Using the formula for foot of perpendicular, we find the coordinates to be (2, 1).

The coordinates of the foot of the perpendicular from the point (1, 2) to the line 3x + 4y = 12 are:

The coordinates of the foot of the perpendicular from the point (1, 2) to the line 3x + 4y = 12 are:

Step 1: Identify the point from which we want to drop a perpendicular. In this case, the point is (1, 2).
Step 2: Write down the equation of the line, which is 3x + 4y = 12.
Step 3: Rearrange the line equation into slope-intercept form (y = mx + b) to find the slope. The line can be rewritten as y = (-3/4)x + 3.
Step 4: Determine the slope of the line, which is -3/4.
Step 5: Find the slope of the perpendicular line. The slope of a line perpendicular to another is the negative reciprocal. So, the slope will be 4/3.
Step 6: Use the point-slope form of the equation of a line to write the equation of the perpendicular line passing through (1, 2). The equation is y - 2 = (4/3)(x - 1).
Step 7: Simplify the equation of the perpendicular line to find its y-intercept form.
Step 8: Solve the two equations (the original line and the perpendicular line) simultaneously to find the intersection point, which is the foot of the perpendicular.
Step 9: After solving, you will find the coordinates of the foot of the perpendicular to be (2, 1).

Foot of Perpendicular – Finding the point on a line that is closest to a given point, which involves using the slope of the line and the negative reciprocal for the perpendicular.
Line Equation – Understanding the standard form of a line equation and how to manipulate it to find points of intersection.
Coordinate Geometry – Applying coordinate geometry principles to solve problems involving distances and angles between points and lines.