The Hyperbola
A ship in the Gulf of Mexico receives the radio signal from two stations on the coast and notices that one reaches it slightly before the other. From that single piece of data it already knows it is on a very specific curve, with two branches that open outward without limit: the hyperbola. How can a difference in times give the ship a curve to locate itself on? We will answer that at the end.
The hyperbola as a locus
Determine the locus of points in the plane whose difference of distances to the fixed points $F(0, 10)$ and $F'(0, -10)$ is, in absolute value, equal to $12$. Write its…
Determine the locus of points in the plane whose difference of distances to the fixed points $F(0, 10)$ and $F'(0, -10)$ is, in absolute value, equal to $12$. Write its standard equation.
Elements and standard equation
If the foci are on the $Y$-axis, the roles are swapped: $\dfrac{y^2}{a^2}-\dfrac{x^2}{b^2}=1$, with vertices $(0, \pm a)$ and foci $(0, \pm c)$. The positive term tells you which way the curve opens.
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 $8$.
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.
Find the standard equation of the hyperbola that has its vertices at the points $V(3, 0)$ and $V'(-3, 0)$ and passes through the point $P(5, 4)$.
Find the standard equation of the hyperbola that has its vertices at the points $V(3, 0)$ and $V'(-3, 0)$ and passes through the point $P(5, 4)$.
Asymptotes and eccentricity
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 $16x^2 - 9y^2 = 144$.
Find the equation of the hyperbola centred at the origin that has a focus at the point $F(0, \sqrt{13})$ and where one of its asymptotes is the line with equation…
Find the equation of the hyperbola centred at the origin that has a focus at the point $F(0, \sqrt{13})$ and where one of its asymptotes is the line with equation $2x - 3y = 0$.
Hyperbola with its center away from the origin
If you are given the general equation $Ax^2+Cy^2+Dx+Ey+F=0$ with $A$ and $C$ of opposite signs, complete the square in $x$ and in $y$ to bring it to standard form.
Find the coordinates of the center, the lengths of the semi-axes, and the equations of the asymptotes of the hyperbola whose equation is…
Find the coordinates of the center, the lengths of the semi-axes, and the equations of the asymptotes of the hyperbola whose equation is $\frac{(x+2)^2}{9} - \frac{(y-1)^2}{16} = 1$.
Given the general equation of a conic section $9x^2 - 4y^2 + 54x + 16y + 29 = 0$, group it by completing the square to prove that it is a hyperbola. Calculate the…
Given the general equation of a conic section $9x^2 - 4y^2 + 54x + 16y + 29 = 0$, group it by completing the square to prove that it is a hyperbola. Calculate the coordinates of its center, its foci and its vertices.
Points and lines relative to the hyperbola
A point $(x_0,y_0)$ is on $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$ if substituting gives $1$. If $\dfrac{x_0^2}{a^2}-\dfrac{y_0^2}{b^2}>1$ it is inside a branch (on the side of the focus); if it is less than $1$, it lies between the branches.
For a line, substitute its equation into that of the hyperbola and analyze the resulting quadratic with the discriminant:
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$\Delta>0$: secant (two points).
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$\Delta=0$: tangent (one point).
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$\Delta<0$: does not intersect.
Check analytically whether the points $A(5, 4)$, $B(4, 0)$ and $C(6, -3)$ lie inside, outside, or on the hyperbola with equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$.
Check analytically whether the points $A(5, 4)$, $B(4, 0)$ and $C(6, -3)$ lie inside, outside, or on the hyperbola with equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$.
Calculate the points of intersection of the line $x - y - 2 = 0$ and the hyperbola $x^2 - 2y^2 = 2$. Next, determine the exact length of the chord formed when the line…
Calculate the points of intersection of the line $x - y - 2 = 0$ and the hyperbola $x^2 - 2y^2 = 2$. Next, determine the exact length of the chord formed when the line intersects the conic section.
Find the equation of the tangent line and the normal line to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $P\left(5, \frac{9}{4}\right)$.
Find the equation of the tangent line and the normal line to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $P\left(5, \frac{9}{4}\right)$.
Determine the values of the parameter $k$ for which the line $y = x + k$ is tangent, secant or external to the hyperbola $x^2 - 2y^2 = 4$.
Determine the values of the parameter $k$ for which the line $y = x + k$ is tangent, secant or external to the hyperbola $x^2 - 2y^2 = 4$.
Find the equations of the tangent lines to the hyperbola $x^2 - 4y^2 = 4$ drawn from the exterior point $P(1, 1)$.
Find the equations of the tangent lines to the hyperbola $x^2 - 4y^2 = 4$ drawn from the exterior point $P(1, 1)$.
Applications: LORAN navigation and more
In the LORAN system, the two stations are at the foci. The ship measures the time difference between the two signals; multiplied by the speed of the wave, it gives $2a$. That constant difference of distances is exactly the definition: the ship is on one branch of a known hyperbola. With a second pair of stations it obtains another hyperbola, and its position is the intersection. That answers the question from the beginning.
Other examples: the cooling towers of a power plant (hyperboloids), the secondary mirror of a Cassegrain telescope, the path of a comet that passes near the Sun only once (the Sun at a focus), and the region lit on a wall by a desk lamp.
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 kilometers. 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 $y = 45\text{ km}$.
A table lamp projects light upward and downward forming a cone of light. When placed near a vertical wall, the boundary of the illuminated region forms a hyperbola.…
A table lamp projects light upward and downward forming a cone of light. When placed near a vertical wall, the boundary of the illuminated region forms a hyperbola. Measuring along the wall and taking the floor as the x-axis, it is observed that the center of the hyperbola is at a height of $80\text{ cm}$. The vertex of the upper branch is at $100\text{ cm}$ from the floor. If at a height of $180\text{ cm}$ from the floor the total width of the illuminated region is $96\text{ cm}$, determine the eccentricity of the projected hyperbola.
An interstellar comet approaches the solar system following a hyperbolic trajectory, with the Sun located at one of its foci. At its closest approach (perihelion), the…
An interstellar comet approaches the solar system following a hyperbolic trajectory, with the Sun located at one of its foci. At its closest approach (perihelion), the comet passes $2\text{ UA}$ (astronomical units) from the Sun. Astronomical observations indicate that the angle formed by the two asymptotes of the comet's trajectory is $60^\circ$. Taking the Sun at the focus with positive abscissa and the focal axis along the $X$-axis (with the vertex to the left of the Sun), determine the equation of the comet's trajectory and the distance at which it will be from the Sun when it crosses the Sun's vertical line.
More challenging problems
For locus problems, take $P(x,y)$, write the condition using the distance or slope formula, and simplify until you recognize the form $\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1$.
Find the locus of points $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(4, 0)$ and $B(-4, 0)$ is equal to…
Find the locus of points $P(x, y)$ such that the product of the gradients of the lines connecting them to the fixed points $A(4, 0)$ and $B(-4, 0)$ is equal to $\frac{9}{16}$. Prove analytically that it is a hyperbola, find its standard equation, the coordinates of its foci and its eccentricity.
Prove analytically that the area of the triangle formed by any tangent line to the rectangular hyperbola with equation $xy = k$ (with $k > 0$) and the two coordinate…
Prove analytically that the area of the triangle formed by any tangent line to the rectangular hyperbola with equation $xy = k$ (with $k > 0$) and the two coordinate axes is independent of the point of tangency. Calculate this constant value in terms of the parameter $k$.
In the design of a Cassegrain reflecting telescope, the primary mirror is parabolic and the secondary mirror has the shape of one branch of a hyperbola. The focus of the…
In the design of a Cassegrain reflecting telescope, the primary mirror is parabolic and the secondary mirror has the shape of one branch of a hyperbola. The focus of the parabolic mirror coincides with one of the foci of the hyperbola, located at $120\text{ cm}$ in front of the vertex of the secondary mirror. The final image is formed at the other focus of the hyperbola, located at $30\text{ cm}$ behind that vertex (that is, the total distance between the two foci is $150\text{ cm}$). Calculate the equation of the hyperbola defining the secondary mirror and its eccentricity, placing the center of the hyperbola at the origin of coordinates and the focal axis along the x-axis.
Study, in terms of the parameter $m$, the relative position of the family of lines $y = mx - 1$ with respect to the rectangular hyperbola $x^2 - y^2 = 1$. Specify for…
Study, in terms of the parameter $m$, the relative position of the family of lines $y = mx - 1$ with respect to the rectangular hyperbola $x^2 - y^2 = 1$. Specify for which values of $m$ the line intersects a single branch, both branches, is tangent, or does not intersect the curve.
Determine the equation of a hyperbola centred at the origin with axes coinciding with the coordinate axes, knowing that it passes through the points $A(4, \sqrt{3})$ and…
Determine the equation of a hyperbola centred at the origin with axes coinciding with the coordinate axes, knowing that it passes through the points $A(4, \sqrt{3})$ and $B(2\sqrt{3}, 1)$. From the equation obtained, calculate the exact distance between its foci.
Let $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be the equation of a hyperbola. Find, in terms of $a$, $b$ and $c$, the coordinates of the points $P$ belonging to the curve…
Let $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$ be the equation of a hyperbola. Find, in terms of $a$, $b$ and $c$, the coordinates of the points $P$ belonging to the curve from which the segment joining the two foci $F$ and $F'$ subtends a right angle (that is, the position vectors $PF$ and $PF'$ are perpendicular). Prove analytically that such points always exist for any hyperbola, regardless of its eccentricity.