Find the point on the line 3 x + 2 y - 5 =0 which is closest to the point (4,5).
Take any point on the line as (x,y)
Now, distance between the point on the line and the given point = d
=> d^2 = (x-4)^2 + (y-5)^2
You can treat this as a function. When d^2 is minimum, then d will be minimum too.
function to minimize = (x-4)^2 + (y-5)^2
But, y can be written in terms of x:
3x + 2y = 5
=> y = (5-3x)/2
So, by minimizing the function
f(x) = (x-4)^2 + ((-3x-5)/2)^2
You will get the required point.
To find the point on the line closest to a given point, we can use the concept of orthogonal projection.
Step 1: Rewrite the equation of the line in slope-intercept form (y = mx + c):
3x + 2y - 5 = 0
2y = -3x + 5
y = (-3/2)x + 5/2
Step 2: Calculate the slope of the given line, which is perpendicular to the line we're interested in finding the closest point on. The slope of the given line is (-2/3).
Step 3: Find the equation of the line perpendicular to the given line and passing through the point (4,5). Since the slopes of perpendicular lines are negative reciprocals, the slope of the new line is (3/2). Using the point-slope form, we can write the equation of the new line as:
y - 5 = (3/2)(x - 4)
Step 4: Now, we will find the intersection point of the two lines by solving the system of equations consisting of the line we are interested in and the new line. Substituting the equation of the new line into the equation of the given line, we get:
(-3/2)x + 5/2 = (3/2)(x - 4)
(-3/2)x + 5/2 = (3/2)x - 6
(-3/2)x - (3/2)x = -6 - 5/2
-3x - 3x = -12 - 5
-6x = -17
x = -17/-6
x = 17/6
Step 5: Substitute the value of x into either equation (given line or new line) to find the corresponding y-coordinate. Using the given line equation:
2y = -3(17/6) + 5
2y = -17/2 + 5
2y = -17/2 + 10/2
2y = -7/2
y = -7/4
So, the point on the line 3x + 2y - 5 = 0 closest to the point (4,5) is (17/6, -7/4).