For some k>0 the lines 50x+ky=1240 and ky=8x+544 intersect at right angles at the point (m,n).find m+n.
To find the value of m + n, we first need to determine the values of m and n.
Let's solve the system of equations formed by the two given lines:
Equation 1: 50x + ky = 1240
Equation 2: ky = 8x + 544
To make it easier to solve, we can rewrite Equation 2 by isolating ky:
ky = 8x + 544
Divide both sides by k:
y = (8/k)x + (544/k)
Now, we need to find the value of k. Since we know that the lines intersect at right angles, we can use the property that the slopes of perpendicular lines are negative reciprocals of each other.
The slope of the line in Equation 1 can be determined by rearranging it into slope-intercept form (y = mx + b):
50x + ky = 1240
ky = -50x + 1240
y = -50/kx + 1240/k
Comparing the above equation with the form y = mx + b, we can see that the slope, m, of the line in Equation 1 is -50/k.
Now, the slope of the line in Equation 2 is 8/k.
Since the slopes are negative reciprocals of each other, we can write the equation:
(-50/k) * (8/k) = -1
Simplifying this equation, we get:
-400/k^2 = -1
Multiply both sides by -1 to make it easier to solve:
400/k^2 = 1
Now, cross-multiply:
k^2 * 1 = 400
This simplifies to:
k^2 = 400
Taking the square root of both sides:
k = ± √400
k can be either 20 or -20.
Let's consider the case where k = 20:
Substituting k = 20 into Equation 1, we have:
50x + 20y = 1240
10x + 4y = 248
Next, we substitute k = 20 into Equation 2 and rewrite it in slope-intercept form:
20y = 8x + 544
y = (8/20)x + (544/20)
y = (2/5)x + 27.2
Now we have a system of two equations:
10x + 4y = 248
y = (2/5)x + 27.2
Solving this system of equations, we find:
x = 16
y = 33.6
Therefore, when k = 20, the lines intersect at the point (16, 33.6).
Finally, we calculate:
m + n = 16 + 33.6 = 49.6
So, when k = 20, m + n = 49.6.
If we consider the case when k = -20, we can go through the same steps and find the values of m and n. However, the final result will be different since the lines will intersect at a different point.