Answer:
![\frac{2x^{2} }{{25} } + \frac{2y^{2}}{{25} } = 1](https://tex.z-dn.net/?f=%5Cfrac%7B2x%5E%7B2%7D%20%7D%7B%7B25%7D%20%7D%20%2B%20%5Cfrac%7B2y%5E%7B2%7D%7D%7B%7B25%7D%20%20%7D%20%3D%201)
Step-by-step explanation:
Since the required hyperbola has its vertex at (5,0), its transverse axis is on the x-axis and center at (0,0).
So, we use the equation in standard form
![\frac{x^{2} }{a^{2} } + \frac{y^{2} }{b^{2} } = 1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E%7B2%7D%20%7D%7Ba%5E%7B2%7D%20%7D%20%2B%20%5Cfrac%7By%5E%7B2%7D%20%7D%7Bb%5E%7B2%7D%20%7D%20%3D%201)
Also, since the path of the space probe gets closer to y = -x, this is the asymptote to the hyperbola.
Our standard asymptote equation is y = ±bx/a taking the negative sign and comparing with y = -x,
-bx/a = -x ⇒ b/a = 1 ⇒ a = b
Also, the coordinate of the vertex (c, 0) = (5, 0) and c² = a² + b²
substituting c = 5 and a = b into the equation, we have
c² = a² + b²
5² = a² + a²
25 = 2a²
a² = 25/2
a = √(25/2)
a = ±5/√2
rationalizing, we have
a = ±5/√2 × √2/√2
a = ±5√2/2
Since a = b, b = ±5√2/2
Inserting a and b into the equation for the hyperbola, we have
![\frac{x^{2} }{a^{2} } + \frac{y^{2} }{b^{2} } = 1\\\frac{x^{2} }{(\frac{5\sqrt{2} }{2} )^{2} } + \frac{y^{2} }{\frac{5\sqrt{2} }{2} ^{2} } = 1\\\frac{x^{2} }{\frac{25 X 2 }{4} } + \frac{y^{2} }{\frac{25X2 }{4} } = 1\\\frac{x^{2} }{\frac{25}{2} } + \frac{y^{2} }{\frac{25}{2} } = 1\\\frac{2x^{2} }{{25} } + \frac{2y^{2}}{{25} } = 1](https://tex.z-dn.net/?f=%5Cfrac%7Bx%5E%7B2%7D%20%7D%7Ba%5E%7B2%7D%20%7D%20%2B%20%5Cfrac%7By%5E%7B2%7D%20%7D%7Bb%5E%7B2%7D%20%7D%20%3D%201%5C%5C%5Cfrac%7Bx%5E%7B2%7D%20%7D%7B%28%5Cfrac%7B5%5Csqrt%7B2%7D%20%7D%7B2%7D%20%29%5E%7B2%7D%20%7D%20%2B%20%5Cfrac%7By%5E%7B2%7D%20%7D%7B%5Cfrac%7B5%5Csqrt%7B2%7D%20%7D%7B2%7D%20%5E%7B2%7D%20%7D%20%3D%201%5C%5C%5Cfrac%7Bx%5E%7B2%7D%20%7D%7B%5Cfrac%7B25%20X%202%20%7D%7B4%7D%20%20%7D%20%2B%20%5Cfrac%7By%5E%7B2%7D%20%7D%7B%5Cfrac%7B25X2%20%7D%7B4%7D%20%7D%20%3D%201%5C%5C%5Cfrac%7Bx%5E%7B2%7D%20%7D%7B%5Cfrac%7B25%7D%7B2%7D%20%7D%20%2B%20%5Cfrac%7By%5E%7B2%7D%20%7D%7B%5Cfrac%7B25%7D%7B2%7D%20%20%7D%20%3D%201%5C%5C%5Cfrac%7B2x%5E%7B2%7D%20%7D%7B%7B25%7D%20%7D%20%2B%20%5Cfrac%7B2y%5E%7B2%7D%7D%7B%7B25%7D%20%20%7D%20%3D%201)
So, the required equation is
![\frac{2x^{2} }{{25} } + \frac{2y^{2}}{{25} } = 1](https://tex.z-dn.net/?f=%5Cfrac%7B2x%5E%7B2%7D%20%7D%7B%7B25%7D%20%7D%20%2B%20%5Cfrac%7B2y%5E%7B2%7D%7D%7B%7B25%7D%20%20%7D%20%3D%201)