Hello
A straight line passes through the point (1,27) and intersects the positive x-axis at the point A and the positive y-axis at the point B.Find the shortest possible distance between A and B..
Pleaase could you help me?
Thank you in advance
y = m x + b
when x = 0, y = B so b=B
y = m x + B
when y = 0, x = A
0 = mA+ B
so
m = -B/A
so
y = -Bx/A + B
when x = 1, y = 27
27 = -B/A + B
27 A = - B + BA
27 A = B (A-1)
B = 27 A/(A-1)
dB/DA = 27 [ (A-1)-A] /(A-1)^2
d^2 = A^2 + B^2
d d^2/dA = 0 for min
= 2 A + 2B dB/DA
so
0 = 2 A + 2B[ 27 [ (A-1)-A] /(A-1)^2
0 = 2A(A-1)^2 - 54 B
0 = 2A(A-1)^2 -54 [ 27 A/(A-1) ]
27 [27/(A-1) ] = (A-1)^2
27^2 = (A-1)^3 = 729
so
A-1 = 9
A = 10 amazing. You take it from there.
oh well curious now
B = 27(10)/9 = 270/9 = 30
d^2 = 10^2 + 30^2 = 1000
d = 10 sqrt(10)
or about 31.6
Of course! To find the shortest possible distance between the points A and B, we need to find the coordinates of A and B first.
Since the line passes through the point (1,27) and intersects the positive x-axis at A, we can assume that the x-coordinate of point A is positive, and the y-coordinate is zero. So, the coordinates of A are (a,0), where a represents the x-coordinate of A.
Similarly, since the line passes through the point (1,27) and intersects the positive y-axis at B, we can assume that the y-coordinate of point B is positive, and the x-coordinate is zero. So, the coordinates of B are (0,b), where b represents the y-coordinate of B.
Now, let's find the equation of the line passing through (1,27). We can use the point-slope form of a linear equation: y - y1 = m(x - x1), where (x1, y1) are the coordinates of a point on the line, and m is the slope of the line.
Since the line passes through (1,27), we have y - 27 = m(x - 1). To find the value of m, we can use another point on the line, which is (0,b). Substituting these values, we get b - 27 = m(0 - 1), which simplifies to b - 27 = -m.
From this equation, we can express m in terms of b: m = - (b - 27). Substituting this value in the equation y - 27 = m(x - 1), we get y - 27 = - (b - 27)(x - 1).
To find the coordinates of A and B, we substitute the appropriate values. For A, we substitute y = 0, and for B, we substitute x = 0.
For point A:
0 - 27 = - (b - 27)(a - 1).
For point B:
b - 27 = - (b - 27)(0 - 1).
Simplifying these equations, we get:
-27 = (27 - b)(a - 1).
b - 27 = (b - 27)(1).
From the second equation, we can see that b - 27 ≠ 0, otherwise, the equation would imply 0 = 0. Therefore, we can divide both sides of the equation by (b - 27), which gives us 1 = 1. This means that there is no unique value for b.
However, if we look at the first equation, -27 = (27 - b)(a - 1), we can see that for b = 27, the equation becomes -27 = 0(a - 1), which simplifies to -27 = 0. Since this is not possible, we can conclude that there is no real solution for a.
Therefore, we cannot find the coordinates of A and B, and without knowing the coordinates, we cannot find the shortest possible distance between A and B.