If real numbers a and b satisfy a^2 + b^2 = 64, what is the distance between the lines ax + by = 1 and ax + by = 417 ?
To find the distance between two parallel lines, we can use the formula:
distance = |c1 - c2| / sqrt(a^2 + b^2)
where (a, b) is the direction vector of both lines, and c1 and c2 are the constants on the right-hand side of the equations of the lines.
In this case, the direction vector (a, b) is the same for both lines, as they are parallel.
Let's find the direction vector (a, b):
From the equations of the lines ax + by = 1 and ax + by = 417, we can see that the direction vector is (a, b) = (a, b) = (1, 0).
Now we need to find the constants c1 and c2.
For the first line ax + by = 1, we can rearrange the equation to get:
by = 1 - ax
y = (1 - ax) / b
On this line, the constant c1 is 1 - ax.
For the second line ax + by = 417, we can rearrange the equation to get:
by = 417 - ax
y = (417 - ax) / b
On this line, the constant c2 is 417 - ax.
Now we can substitute the values of the constants c1 and c2, and the direction vector (a, b) into the distance formula:
distance = |c1 - c2| / sqrt(a^2 + b^2)
= |(1 - ax) - (417 - ax)| / sqrt(1^2 + 0^2)
= |1 - ax - 417 + ax| / sqrt(1)
= | -416 | / 1
= 416
So, the distance between the lines ax + by = 1 and ax + by = 417 is 416 units.