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The physicist Ernest Rutherford discovered that when alpha particles are directed towards the nuclei of gold atoms, they are eventually deflected along hyperbolic paths.

If a particle can get as close as $8$8 units to the nucleus along a hyperbolic path with an asymptote given by $y=\frac{1}{4}x$`y`=14`x`, what is the equation of its path?

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Easy

Approx 5 minutes

An astronomer is studying the remains of an old star that has ejected its outer atmosphere in bursts of material. A cross section of the nebula has the shape of a hyperbola as plotted below. The units are given in light years. The point $Q$`Q`$\left(0.6,0.2\right)$(0.6,0.2) is on the asymptote of the gas shells. The point $V$`V`$\left(1.2,0\right)$(1.2,0) is the vertex of one of the gas shells.

Comets around the sun sometimes have a hyperbolic orbit. One such comet had a path that could be approximately described by the equation $25x^2-y^2=25$25`x`2−`y`2=25.

Units are measured in Astronomical units (AU), the distance from the Earth to the Sun.

Two stationary communication beacons $55317$55317 m apart, send out a transmission to a ship at the same time. The ship receives the transmissions $120$120 microseconds apart. The ship can use this information to determine where it might be located using LORAN (LOng RAnge Navigation). The possible locations lie along a hyperbola.

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