Find the point on the line –3x+4y–5=0 which is closest to the point (0–5)
the slope of your line is 3/4
So, the slope of the perpendicular line is -4/3
The line through (0,-5) with slope -4/3 is
y+5 = -4/3 x
The two lines intersect at (-3,-1)
Or, if you want to exercise your calculus, the distance from (0,-5) to any point (x,y) is
d^2 = x^2 + (y+5)^2
Since y = (3x+5)/4,
d^2 = x^2 + ((3x+5)/4+5)^2
= 25/16 (x^2+6x+25)
2d dd/dx = 25/16 (2x+6)
dd/dx=0 when x = -3, as above.
To find the point on the line –3x + 4y – 5 = 0 that is closest to the point (0, –5), you can use the concept of perpendicular distance.
Step 1: Convert the given equation of the line into the slope-intercept form (y = mx + b).
-3x + 4y – 5 = 0
4y = 3x + 5
y = (3/4)x + 5/4
Step 2: Calculate the slope of the given line. In this case, the slope is 3/4.
Step 3: Find the perpendicular line passing through the point (0, –5). Since the line to be found is perpendicular to the given line, its slope will be the negative reciprocal of the slope of the given line. In this case, the slope of the perpendicular line will be –4/3.
Step 4: Use the point-slope form of a linear equation to find the equation of the perpendicular line.
(y - y1) = m(x - x1)
(y - (-5)) = (-4/3)(x - 0)
y + 5 = (-4/3)x
Simplifying further, we get:
y = (-4/3)x - 5
Step 5: Solve the system of equations formed by the given line and the perpendicular line to find the point of intersection.
(3/4)x + (5/4) = (-4/3)x - 5
Multiplying both sides of the equation by 12 to eliminate the fractions, we get:
9x + 15 = -16x - 60
Combining like terms, we have:
25x = -75
x = -3
Substituting the value of x back into either equation, we find:
y = (3/4)(-3) + 5/4
y = -9/4 + 5/4
y = -4/4
y = -1
Therefore, the point on the line –3x + 4y – 5 = 0 that is closest to the point (0, –5) is (-3, -1).