Question: The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:
Options:
(2, 0)
(0, 2)
(1, 1)
(2, 2)
Correct Answer: (2, 0)
Solution:
Using the formula for foot of perpendicular, we find the coordinates to be (2, 0).
The coordinates of the foot of the perpendicular from the point (3, 4) to the li
Practice Questions
Q1
The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:
(2, 0)
(0, 2)
(1, 1)
(2, 2)
Questions & Step-by-Step Solutions
The coordinates of the foot of the perpendicular from the point (3, 4) to the line 2x + 3y - 6 = 0 are:
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).
Foot of the 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 the standard form of a line equation and how to manipulate it to find perpendicular distances.
Coordinate Geometry – Applying principles of coordinate geometry to solve problems involving points and lines in a Cartesian plane.
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