Solve the problem.
-A system for tracking ships indicated that a ship lies on a hyperbolic path described by $5 x ^ { 2 } - y ^ { 2 } = 31$. The process is repeated and the ship is found to lie on a hyperbolic path described by $y ^ { 2 } - 3 x ^ { 2 } = 1$. If it is known that the ship is located in the first quadrant of the coordinate system, determine its exact location.
A) $( 4,7 )$
B) $( - 4 , - 7 )$
C) $( 7,4 )$
D) $( - 7 , - 4 )$

Question

Quizplus · Accepted Answer

To find the ship's location, we solve the system of equations given by the two hyperbolic paths. The equations are:1. $5x^2 - y^2 = 31$2. $y^2 - 3x^2 = 1$Adding these equations eliminates $y^2$, resulting in:$5x^2 - y^2 + y^2 - 3x^2 = 31 + 1$$2x^2 = 32$$x^2 = 16$$x = 4$ (since the ship is in the first quadrant, we only consider the positive root).Substituting $x = 4$ into the second equation:$y^2 - 3(4)^2 = 1$$y^2 - 48 = 1$$y^2 = 49$$y = 7$ (again, considering the positive root for the first quadrant).Therefore, the ship's exact location is $(4, 7)$.

Solve the Problem 5x2−y2=315 x ^ { 2 } - y ^ { 2 } = 315x2−y2=31

Solve the Problem $5 x ^ { 2 } - y ^ { 2 } = 31$