The coordinates of the foot of the perpendicular from the point (1, 2) to the li
Practice Questions
Q1
The coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x + 3y = 6 are:
(2, 0)
(0, 2)
(1, 1)
(0, 0)
Questions & Step-by-Step Solutions
The coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x + 3y = 6 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 to which we are dropping the perpendicular. The line is given as 2x + 3y = 6.
Step 3: Rearrange the line equation into slope-intercept form (y = mx + b) to find the slope. The rearranged form is y = -2/3 x + 2.
Step 4: Determine the slope of the line. From the equation y = -2/3 x + 2, 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 to find its y-intercept form. This gives us y = (3/2)x - (3/2) + 2, which simplifies to y = (3/2)x + 1/2.
Step 8: Now, we have two equations: the original line (2x + 3y = 6) 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.
Step 11: After solving, you will find x = 2.
Step 12: Substitute x = 2 back into the equation of the perpendicular line to find the corresponding y-coordinate.
Step 13: After substituting, you will find y = 0.
Step 14: Therefore, the coordinates of the foot of the perpendicular from the point (1, 2) to the line 2x + 3y = 6 are (2, 0).
