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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 ?

CrabSis
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ProblemDestroyer
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1 Answers1

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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 $(-6,-8), (-3,-4), (0,0), (3,4), (6,8)$.

CrabSis
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  • Sir thanks i understood , but in general is there a way to find all the integral points on a hyperbola constrained between some x -range? – ProblemDestroyer Feb 25 '22 at 10:55
  • I don't know if there is a general method, but I believe that this method should apply for $|\sqrt{(x-x_1)^2+(y-y_1)^2}-\sqrt{(x-x_2)^2+(y-y_2)^2}|=\sqrt{(x_1-x_2)^2+(y_1-y_2)^2}$. – CrabSis Feb 25 '22 at 11:17
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    Hmm understood this was easily solved due to degenerate hyperbola reducing it to a part of a straight line. – ProblemDestroyer Feb 25 '22 at 11:36