Line through (0,b) and (10,-5) is perpendicular to the line through (a,0) and (-6,-3). If all ordered pairs (a,b) which make the first sentence of this problem true are graphed, they all lie on a line. Find the equation of this line.
We have two lines, with slopes
(-5-b)/10 and (-3)/(-6-a)
If the lines are perpendicular, then
(-5-b)/10 * (-3)/(-6-a) = -1
(b+5) = 10/3 (a+6)
b = 5/3 (2a+9)
To find the equation of the line that all the points (a, b) lie on, we need to find the relationship between a and b.
Let's start by finding the slope of the line through (0, b) and (10, -5). The formula for the slope of a line passing through two points (x1, y1) and (x2, y2) is given by:
m = (y2 - y1) / (x2 - x1)
In our case, the points are (0, b) and (10, -5). Plugging in these values, we get:
m1 = (-5 - b) / (10 - 0)
m1 = (-5 - b) / 10
Now, let's find the slope of the line through (a, 0) and (-6, -3). Using the same formula, we have:
m = (-3 - 0) / (-6 - a)
m = -3 / (-6 - a)
m = 1/2 + a/6
Since the slope of two perpendicular lines is the negative reciprocal of each other, we can set the two slopes equal to each other and solve for a:
(-5 - b) / 10 = -1 / (1/2 + a/6)
Now, let's simplify and solve for a:
(-5 - b) / 10 = -1 / (1/2 + a/6)
(-5 - b) / 10 = -1 / (1/2 + a/6) × (2/2)
(-5 - b) / 10 = -2 / (1 + a/3)
(-5 - b) / 10 = -2 / ((3 + a)/3)
(-5 - b) / 10 = -6 / (3 + a)
-5 - b = -6 / (3 + a) × 10
-5 - b = -60 / (3 + a)
-5(3 + a) - b(3 + a) = -60
-15 - 5a - 3b - ab = -60
-15 - 3b - 5a - ab = -60
-15 - 3b - 5a - ab + 60 = 0
45 - 3b - 5a - ab = 0
Now we have the equation of the line:
45 - 3b - 5a - ab = 0
This is the equation of the line that all points (a, b) satisfying the conditions of the problem lie on.