Let 5a+12b and 12a+5b be the side lengths of a right-angled triangle and 13a+kb be the hypotenuse, where a,b and k are positive integers. find the smallest possible value of k and the smallest values of a and b for that k.
To find the smallest value of k and the corresponding values of a and b, we need to solve the given equations and find the values that satisfy the conditions of a right-angled triangle.
We are given the side lengths of the right-angled triangle as 5a+12b and 12a+5b, and the hypotenuse as 13a+kb.
According to the Pythagorean theorem, in a right-angled triangle, the sum of the squares of the two shorter sides is equal to the square of the hypotenuse.
Using this information, we can set up the following equation:
(5a+12b)^2 + (12a+5b)^2 = (13a+kb)^2
Expanding the equation:
25a^2 + 120ab + 144b^2 + 144a^2 + 60ab + 25b^2 = 169a^2 + 26akb + k^2b^2
Combining like terms and simplifying:
169a^2 + 25b^2 - k^2b^2 - 240ab - 26akb - 169a^2 - 25b^2 = 0
240ab + 26akb = 0
Factoring out the common factors:
ab (240 + 26k) = 0
For this equation to hold true, either ab must be equal to 0, or the quantity (240 + 26k) must be equal to 0.
Since a and b are positive integers, ab cannot be equal to 0. Therefore, we need to find the values of k for which (240 + 26k) equals 0.
Solving the equation (240 + 26k) = 0:
26k = -240
k = -240 / 26
k ≈ -9.2308
Since k needs to be a positive integer, we need to round up to the nearest positive integer.
The smallest value of k is 10.
To find the corresponding values of a and b, substitute k = 10 into one of the original two equations:
5a + 12b = 13a + 10b
Rearranging the equation:
-8a = -2b
a = b / 4
To find the smallest values of a and b, we can try different positive integer values for b and calculate the corresponding values of a:
If b = 4, then a = 1 (smallest values)
Therefore, the smallest possible value of k is 10, and the corresponding smallest values of a and b are a = 1 and b = 4, respectively.