Evidence for the distance of the point on the isosceles hyperbola to the center?

[gunza]

The product of the distances of any point P(x, y) to the foci of H on an isosceles hyperbola given by the equation [MATH]{H:x^{2} - y^{2} = a^{2}}[/MATH] is equal to the square of the distance P to the center of H.

Is there evidence of this?

 

[Deleted member 4993]

The product of the distances of any point P(x, y) to the foci of H on an isosceles hyperbola given by the equation [MATH]{H:x^{2} - y^{2} = a^{2}}[/MATH] is equal to the square of the distance P to the center of H.

Is there evidence of this?

Please show us what you have tried and exactly where you are stuck.

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[gunza]

This is a evidence question (from the analytical geometry project). I searched but couldn't find the proof.
Isosceles hyperbola equation: [MATH]{H = x^{2}-y^{2} = a^{2}}[/MATH]And let's take any point P(x, y) on this hyperbola. Now, the product of the distances of this point P(x, y) to the foci of the hyperbola is equal to the square of the distance from point P to the center of the hyperbola.

This is the claim and I'm looking for evidence of it. I hope I could explain.Thanks for your feedback.

 

[Dr.Peterson]

The product of the distances of any point P(x, y) to the foci of H on an isosceles hyperbola given by the equation [MATH]{H:x^{2} - y^{2} = a^{2}}[/MATH] is equal to the square of the distance P to the center of H.

Is there evidence of this?

To find some evidence, pick a point and try it.

Do you know where the foci are?