Answer:
To find the values of a and b in the equation x + by - 12 = 0 for the given line I passing through (7a, 5) and (3a, 3), we can use the slope-intercept form of a line equation: y = mx + c, where m is the slope and c is the y-intercept.
First, let's find the slope of the line I:
The slope (m) can be found using the formula:
m = (y2 - y1) / (x2 - x1)
Using the coordinates (7a, 5) and (3a, 3):
m = (3 - 5) / (3a - 7a)
= -2 / (-4a)
= 1 / (2a)
Now, we know the slope (m) and one point (3a, 3) on the line I. We can substitute these values into the slope-intercept form to get the equation:
y = mx + c
3 = (1 / (2a))(3a) + c
3 = 3/2 + c
c = 3 - 3/2
c = 3/2
So, the equation of line I in slope-intercept form is y = (1 / (2a))x + 3/2.
Now, we need to compare this equation with x + by - 12 = 0 to find the value of b:
Comparing the coefficients:
1 / (2a) = 1, which implies 2a = 1, then a = 1/2.
b = -1, as the coefficient of y is -1.
Therefore, the value of a is 1/2 and the value of b is -1.