Is the pythagorean triplets idea sufficient to solve this problem and is there another way to get the integer points on hyperbola?

Question

Number of integral points inside circle $x^2 + y^2 = 117$ and satisfying the equation $|\sqrt{x^2+y^2} -\sqrt{(x-3)^2 + (y-4)^2}|= 5$ is :
What i did was using the fact that as we want integral values so the square roots in the equation which needs to be satisfied must both have integral values , so smallest possible pair of triplets would be 3,4,5 and also 6,8,10 will work but will it count all the possible integral values or something more needs to be checked ? And is there another way to find integral points on that hyperbola equation which the points inside the circle are satisfying ?

Answer 1

Recall that the distance formula is $\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$, so essentially, the question is asking for points satisfying $x^2+y^2≤117$ and the point $(x,y)$ is 5 units closer to $(3,4)$ than the origin, or the other way around.

To find this, we can set up a triangle with its three vertices A,B,C at (0,0), (3,4), (x,y) respectively. If we let $AC=x$, then $BC$ is either $x+5$ or $x-5$. Substituting the values into the triangle inequality, we would find out that ABC needs to be colinear. From here, it could be seen that the only valid points are $(-6,-8), (-3,-4), (0,0), (3,4), (6,8)$.