Hyperbola
A ship in the middle of the sea picks up radio signals from two coastal stations and notices that one reaches it before the other: that is enough to know that it lies on a very specific curve, with two branches opening out towards infinity. It is the hyperbola, the conic used by LORAN navigation systems and by the cooling towers of thermal power stations.
The hyperbola as a locus
Write the standard equation of the hyperbola whose foci are $F(5, 0)$ and $F'(-5, 0)$ and whose transverse axis measures $8$.
Write the standard equation of the hyperbola whose foci are $F(5, 0)$ and $F'(-5, 0)$ and whose transverse axis measures $8$.
Elements and standard equation
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine the coordinates of its vertices, of its foci, and the equations of its asymptotes.
Given the hyperbola with equation $\frac{x^2}{144} - \frac{y^2}{25} = 1$, determine the coordinates of its vertices, of its foci, and the equations of its asymptotes.
The asymptotes
Determine the equation of a rectangular hyperbola centred at the origin, knowing that it passes through the point $P(4, 2)$ and that its foci lie on the x-axis.
Determine the equation of a rectangular hyperbola centred at the origin, knowing that it passes through the point $P(4, 2)$ and that its foci lie on the x-axis.
Calculate the eccentricity of the hyperbola whose general equation is given by the expression $16x^2 - 9y^2 = 144$.
Calculate the eccentricity of the hyperbola whose general equation is given by the expression $16x^2 - 9y^2 = 144$.
Eccentricity and LORAN navigation
The LORAN system places two radio stations at the foci; a ship measures the time difference with which it receives both signals, and that difference (multiplied by the speed of the wave) is directly $2a$: the ship is on a known branch of a hyperbola.
In the design of a cooling tower for a thermal power station, the structure has the shape of a hyperboloid of revolution. The vertical cross-section is a hyperbola…
In the design of a cooling tower for a thermal power station, the structure has the shape of a hyperboloid of revolution. The vertical cross-section is a hyperbola centred at the origin of coordinates. The tower has a total height of $160\text{ m}$. The narrowest part (the throat) has a diameter of $60\text{ m}$ and is located at a height of $120\text{ m}$ above the base. Furthermore, the diameter at the top of the tower is $100\text{ m}$. Determine the equation of the hyperbola outlining the tower and calculate the diameter of the circular base.
The LORAN navigation system determines the position of a ship by measuring the time difference in receiving signals from two radio stations. The stations $A$ and $B$ are…
The LORAN navigation system determines the position of a ship by measuring the time difference in receiving signals from two radio stations. The stations $A$ and $B$ are located in a Cartesian coordinate system at the points $(-60, 0)$ and $(60, 0)$, with distances expressed in kilometres. A ship $P$ receives the signal from $A$ later than from $B$, indicating that the ship is $100\text{ km}$ closer to $B$ than to $A$. Determine the equation of the hyperbola along which the ship is sailing and calculate the distance from the ship to station $B$ if its y-coordinate is known to be $y = 45\text{ km}$.