 New Zealand
Level 8 - NCEA Level 3

Applications of hyperbolas

Interactive practice questions

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=14x, what is the equation of its path?

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$25x2y2=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.

Outcomes

M8-1

Apply the geometry of conic sections

M8-7

Form and use trigonometric, polynomial, and other non-linear equations

91573

Apply the geometry of conic sections in solving problems

91575

Apply trigonometric methods in solving problems