Question

(a) How many ordered pairs (x,y) of integers are there such that \sqrt{x^2 + y^2} = 5? Does the question have a geometric interpretation?

(b) How many ordered triples (x,y,z) of integers are there such that \sqrt{x^2 + y^2 + z^2} = 7? Does the question have a geometric interpretation?

Answer 1

i need help

Answer 2

Geometric interpretations:\[x^2+y^2=r^2\]
is the equation the circle with radius \(r\) centered at the origin.

Similarly,
\[x^2+y^2+z^2=\rho^2\]
is the equation of the sphere with radius \(\rho\) centered at the origin.

Answer 3

As for finding the integer solutions... Not sure if there's an algebraic way to do it that doesn't involve fixing one (or two) of the variables and seeing if the remaining one is an integer.

For the circle, you can determine there are 8 integer solutions if you are familiar with the Pythagorean triple (3,4,5), i.e. \(3^2+4^2=5^2\).
\[\begin{matrix}
x=3&&y=4\\
x=-3&&y=4\\
x=3&&y=-4\\
x=-3&&y=-4\\
x=4&&y=3\\
x=-4&&y=3\\
x=4&&y=-3\\
x=-4&&y=-3\end{matrix}\]
I don't think there are any more than that, but I'm not certain.

Answer 4

For the sphere, you could try the same approach and look up Pythagorean quadruples.

Answer 5

wow you r good thanks

Answer 6

what about the one with x,y, and z