Find the length of the perpendicular drawn from the point(4,-3) upon the straight line
6x + 3y - 10 = 0. The angle between the axes being 60°.
To find the length of the perpendicular drawn from a point onto a line, we need to follow these steps:
Step 1: Find the slope of the given straight line.
Step 2: Convert the equation of the line into slope-intercept form (y = mx + c) by solving for y.
Step 3: Find the slope of the line perpendicular to the given line. Remember that the product of the slopes of two perpendicular lines is always -1.
Step 4: Find the equation of the line passing through the given point with the perpendicular slope.
Step 5: Find the point of intersection between the given line and the line passing through the given point.
Step 6: Calculate the distance between the given point and the point of intersection, which will be the length of the perpendicular.
Let's go through these steps to find the length of the perpendicular:
Step 1: Find the slope of the given straight line:
The given equation is 6x + 3y - 10 = 0. Let's solve for y:
3y = -6x + 10
y = (-6/3)x + (10/3)
y = -2x + (10/3)
The slope-intercept form of the line is y = -2x + (10/3). The slope of this line is -2.
Step 2: Find the slope of the line perpendicular to the given line:
The product of the slopes of two perpendicular lines is always -1. So, the slope of the line perpendicular to the given line with slope -2 is 1/2.
Step 3: Find the equation of the line passing through the given point with the perpendicular slope:
Using the point-slope form of a line, the equation of the line passing through the point (4, -3) with a slope of 1/2 is:
y - (-3) = (1/2)(x - 4)
y + 3 = (1/2)x - 2
y = (1/2)x - 5
Step 4: Find the point of intersection between the given line and the line passing through the given point:
Solve the system of equations formed by the given line and the line passing through the point (4, -3):
6x + 3y - 10 = 0
y = (1/2)x - 5
Substitute the second equation into the first equation:
6x + 3((1/2)x - 5) - 10 = 0
6x + (3/2)x - 15 - 10 = 0
(15/2)x - 25 = 0
(15/2)x = 25
(1/2)x = 5/3
x = 10/3
Substitute the value of x back into the second equation to find y:
y = (1/2)(10/3) - 5
y = 5/3 - 5
y = -10/3
So, the point of intersection is (10/3, -10/3).
Step 5: Calculate the distance between the given point (4, -3) and the point of intersection (10/3, -10/3):
The formula to calculate the distance between two points (x1, y1) and (x2, y2) is given by:
distance = sqrt((x2 - x1)^2 + (y2 - y1)^2)
Using this formula, we have:
distance = sqrt(((10/3) - 4)^2 + ((-10/3) - (-3))^2)
distance = sqrt((-2/3)^2 + (-1/3)^2)
distance = sqrt(4/9 + 1/9)
distance = sqrt(5/9)
distance = sqrt(5)/3
Therefore, the length of the perpendicular drawn from the point (4, -3) upon the straight line 6x + 3y - 10 = 0 is sqrt(5)/3.