A Coast Guard cutter in the Gulf of Mexico picks up radio pulses from two shore stations. The pulse from one station arrives a little later than the pulse from the other. That small time gap is enough to place the ship on one very specific curve: two open branches that stretch off forever. That curve is the hyperbola, the shape behind LORAN navigation and behind the curved sides of power-plant cooling towers. By the end of this page you will be able to write the equation of the ship's curve.
The hyperbola as a locus
Compare with the ellipse: there the sum of the distances was constant. Here it is the difference. Watch the moving point $P$ below: the two segments change length, but their difference never does.
Every hyperbola has the same set of key parts. Put the center at the origin and the foci on the $x$-axis at $F_1(-c, 0)$ and $F_2(c, 0)$.
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Determine the locus of the 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…
Determine the locus of the 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.
To write the equation you need only two numbers, $a^2$ and $b^2$, plus the orientation.
When you don't know $a^2$ or $b^2$ but you know two points, treat $u = \dfrac{1}{a^2}$ and $v = \dfrac{1}{b^2}$ as the unknowns. The equation $u x^2 - v y^2 = 1$ is linear in $u$ and $v$, so two points give a $2 \times 2$ linear system.
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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 $8$.
Determine the equation of a hyperbola with its center at the origin and axes coinciding with the coordinate axes, knowing that it passes through the points…
Determine the equation of a hyperbola with its center at the origin and 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 obtained equation, calculate the exact distance between its foci.
As you move far out along a branch, the curve straightens out and hugs two lines through the center. Those lines are the asymptotes.
To sketch a hyperbola quickly: plot the center, draw the central rectangle, draw its diagonals (extended), then draw each branch starting at a vertex and bending toward the asymptotes. Try the sliders: the hyperbola's opening depends only on the ratio $b/a$.
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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.
Determine the equation of a rectangular hyperbola centered at the origin knowing that it passes through the point $P(4, 2)$ and that its foci are located on the x-axis.
Determine the equation of a rectangular hyperbola centered at the origin knowing that it passes through the point $P(4, 2)$ and that its foci are located on the x-axis.
Find the equation of the hyperbola centered at the origin that has a focus at the point $F(0, \sqrt{13})$ and in which one of its asymptotes is the line with equation…
Find the equation of the hyperbola centered at the origin that has a focus at the point $F(0, \sqrt{13})$ and in which one of its asymptotes is the line with equation $2x - 3y = 0$.
The right triangle with legs $a$ and $b$ and hypotenuse $c$ connects everything: the asymptote makes an angle $\theta$ with the transverse axis, and
$$\tan\theta = \frac{b}{a}, \qquad e = \frac{c}{a} = \frac{1}{\cos\theta}.$$
Two more facts you will need:
The distance from a vertex to the nearer focus is $c - a$.
Two lines make two angles. The angle $2\theta$ between the asymptotes that contains the branches satisfies $\tan\theta = \dfrac{b}{a}$. Read each problem carefully to see which of the two angles it means.
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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 $16x^2 - 9y^2 = 144$.
A table lamp projects light upwards and downwards forming a luminous cone. When placed near a vertical wall, the edge of the illuminated area forms a hyperbola.…
A table lamp projects light upwards and downwards forming a luminous cone. When placed near a vertical wall, the edge of the illuminated area forms a hyperbola. Measuring on 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 $100\text{ cm}$ from the floor. If at a height of $180\text{ cm}$ from the floor the total width of the illuminated area 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 at $2\text{ UA}$ (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 a positive x-coordinate and the focal axis on the $X$-axis (with the vertex to the left of the Sun), determine the equation of the comet's trajectory and the distance it will be from the Sun when it crosses the y-axis.
Shift the whole picture $h$ units right and $k$ units up: replace $x$ by $x - h$ and $y$ by $y - k$.
Drag the sliders to see the center move while the shape stays the same.
When the equation arrives in general form, complete the square in $x$ and in $y$.
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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 $9x^2 - 4y^2 + 54x + 16y + 29 = 0$, group it by completing the square to prove that it is a hyperbola. Calculate the coordinates of…
Given the general equation of a conic $9x^2 - 4y^2 + 54x + 16y + 29 = 0$, 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: inside, outside, and intersections
Where is a point?
The curve splits the plane into three kinds of places: points on the curve, points inside (in the region of a branch that contains a focus), and points outside (between the branches). Evaluate the left side of the equation at the point.
A line meeting a hyperbola
To find the intersection points of a line and a hyperbola, substitute the line into the hyperbola's equation and solve the resulting equation.
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Check analytically whether the points $A(5, 4)$, $B(4, 0)$, and $C(6, -3)$ are interior, exterior, or if they belong to the hyperbola with equation…
Check analytically whether the points $A(5, 4)$, $B(4, 0)$, and $C(6, -3)$ are interior, exterior, or if they belong to 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 $x^2 - 2y^2 = 2$. Then, determine the exact length of the chord that bounds the line…
Calculate the points of intersection of the line $x - y - 2 = 0$ and the hyperbola $x^2 - 2y^2 = 2$. Then, determine the exact length of the chord that bounds the line when intersecting the conic.
The tangent at $P$ touches the hyperbola only at $P$. The normal at $P$ is the line through $P$ perpendicular to the tangent, so its slope is the negative reciprocal of the tangent's slope.
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Find the equation of the tangent line and of 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 of the normal line to the hyperbola $\frac{x^2}{16} - \frac{y^2}{9} = 1$ at the point $P\left(5, \frac{9}{4}\right)$.
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.
The LORAN system puts two radio stations at the foci. A ship measures the time difference between the two signals. Multiplying by the signal speed gives the difference of distances, which is exactly $2a$. So the ship is on one branch of a known hyperbola. A second pair of stations gives a second hyperbola, and the ship is where the two curves cross.
Back to the start: the Coast Guard ship sits on the hyperbola whose foci are the two stations and whose $2a$ is the signal-speed times time-difference.
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In the design of a cooling tower for a power plant, the structure has the shape of a hyperboloid of revolution. The vertical cross-section is a hyperbola with its center…
In the design of a cooling tower for a power plant, the structure has the shape of a hyperboloid of revolution. The vertical cross-section is a hyperbola with its center 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 highest part of the tower is $100\text{ m}$. Determine the equation of the hyperbola that outlines 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 the reception of signals from two radio stations. The stations $A$ and…
The LORAN navigation system determines the position of a ship by measuring the time difference in the reception of 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 miles. A ship $P$ receives the signal from $A$ a time after that from $B$, which indicates that the ship is $100\text{ mi}$ closer to $B$ than to $A$. Determine the equation of the hyperbola on which the ship is sailing and calculate the distance of the ship to the station $B$ if it is known that its y-coordinate is $y = 45\text{ mi}$.
In the design of a Cassegrain reflector telescope, the primary mirror is parabolic and the secondary mirror has the shape of a branch of a hyperbola. The focus of the…
In the design of a Cassegrain reflector telescope, the primary mirror is parabolic and the secondary mirror has the shape of a 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 this vertex (i.e., the total distance between the two foci is $150\text{ cm}$). Calculate the equation of the hyperbola that defines the secondary mirror and its eccentricity, placing the center of the hyperbola at the origin of coordinates and the focal axis on the x-axis.
Sometimes a problem hands you a geometric condition and asks which curve it describes. The method never changes: call the moving point $P(x, y)$, translate the condition into an equation in $x$ and $y$, simplify, and recognize the curve.
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Find the locus of the points $P(x, y)$ such that the product of the slopes of the lines connecting them to the fixed points $A(4, 0)$ and $B(-4, 0)$ is equal to…
Find the locus of the points $P(x, y)$ such that the product of the slopes 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 reduced 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 $xy = k$ (where $k > 0$) and the two coordinate…
Prove analytically that the area of the triangle formed by any tangent line to the rectangular hyperbola with equation $xy = k$ (where $k > 0$) and the two coordinate axes is independent of the point of tangency. Calculate this constant value in terms of the parameter $k$.
Let the hyperbola have the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Find, in terms of $a$, $b$, and $c$, the coordinates of the points $P$ belonging to the…
Let the hyperbola have the equation $\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1$. Find, in terms of $a$, $b$, and $c$, the coordinates of the points $P$ belonging to the curve from which the line segment joining the two foci $F$ and $F'$ subtends a right angle (i.e., the focal radii $PF$ and $PF'$ are perpendicular). Prove analytically that such points always exist for any hyperbola, regardless of its eccentricity.