Let P be a point on hyperbola x^2 - y^2 = 1 and Q be a point on hyperbola y^2 - x^2 = 1, where P and Q lie with in a circle x^2 + y^2 \leq 17. What is the least possible distance between P and Q?
I am assuming the last equation is:
x^2 + y^2 < 17
Since the hyperbolas become closer and closer the further out they get, the least possible distance would be the intersections on the circle. There are 2 intersections in each quadrant. However, we can assume the distances between all intersections are equal because of symmetry. Using the first quadrant and a TI graphing calculator to determine the intersections, the intersections are:
(3,sqrt8) and (sqrt8,3)
Using the distance formula to find the distance, we get:
d = sqrt((3 - sqrt8)^2 + (sqrt8 - 3)^2) = 0.24 units
The circle with radius root(17) and a hyperbola will have 1 intersection in each quadrant.
By that logic, the circle has 2 intersections in each quadrant. We can compare a point from one quadrant to another, and the distance will be much greater than when we compare the intersections inside one quadrant.