A line passes through the origin and through points A(−2, b−14) and B(14−b, 72). What is the greatest possible value of b?
the slope is constant
(b - 14) / -2 = 72 / (14 - b) ... solve for b
also, there is indeed a y-intercept. It just happens to be zero.
To find the greatest possible value of b for the given line, we need to find the equation of the line passing through the origin and points A and B.
The equation of a line passing through two points, (x₁, y₁) and (x₂, y₂), is given by:
(y - y₁) = ((y₂ - y₁) / (x₂ - x₁)) * (x - x₁)
Let's substitute the coordinates of points A and B into the equation:
For point A: (x₁, y₁) = (-2, b - 14)
For point B: (x₂, y₂) = (14 - b, 72)
The equation becomes:
(y - (b-14)) = ((72 - (b-14)) / ((14 - b) - (-2))) * (x - (-2))
Simplifying further:
(y - (b-14)) = ((72 - b + 14) / (14 - b + 2)) * (x + 2)
(y - (b-14)) = ((86 - b) / (16 - b)) * (x + 2)
Since the line passes through the origin (0, 0), we can substitute x = 0 and y = 0 into the equation to find a relationship between b, x, and y:
0 - (b-14) = ((86 - b) / (16 - b)) * (0 + 2)
-(b - 14) = ((86 - b) / (16 - b)) * 2
Simplifying further:
-2b + 28 = (172 - 2b) / (16 - b)
To find the greatest possible value of b, we can use the fact that "b" cannot be equal to 16, which would result in division by zero.
Multiplying both sides of the equation by (16 - b) to eliminate the denominator:
-2b(16 - b) + 28(16 - b) = 172 - 2b
-32b + 2b² + 448 - 28b = 172 - 2b
2b² - 2b - 28b + 2b - 32b + 172 - 448 = 0
2b² - 62b - 276 = 0
Now, we can solve this quadratic equation to find the possible values of b:
The greatest possible value of b will give the highest value for the y-coordinate of point A.
Since it passes through the origin, there is no y-intercept
and the equation has the form
y = mx
(−2, b−14) and B(14−b, 72)
slope = (72 - b + 14)/(14-b + 2) = (86-b)/(16-b)
the "critical values of" seem to be 86 and 16
If b > 86, the slope is positive
if b < 16, the slope is positive
for 16 < b < 86 the slope is negative
There doesn't seem to be a greatest value of b.
e.g. let b = 1000
then the slope is (86-1000)/(16-1000) = 457/492 = .9288..
and your equation is y = 457/492 x
as x ---> ∞ , slope ----> 1
Go with R_scott's answer.
I was overthinking the problem.