Exercises: Hyperbola

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.

Check analytically whether the points $A(5, 4)$, $B(4, 0)$ and $C(6, -3)$ lie inside, outside, or on the hyperbola with equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$.

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 $16x^2 - 9y^2 = 144$.

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 $P(5, 4)$.

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.

Find the coordinates of the centre, the lengths of the semi-axes, and the equations of the asymptotes of the hyperbola whose equation is $\frac{(x+2)^2}{9} - \frac{(y-1)^2}{16} = 1$.

Lucas attempts to calculate the foci, eccentricity, and asymptotes of the hyperbola $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$. In his notebook, he writes the following solution:

"I identify $a^2 = 16$ and $b^2 = 9$. Since the fundamental relationship in conics is $a^2 = b^2 + c^2$, I solve for $c = \sqrt{16 - 9} = \sqrt{7} \approx 2,65$. Therefore, the foci are at $(\pm\sqrt{7}, 0)$ and the eccentricity is $e = \dfrac{c}{a} = \dfrac{\sqrt{7}}{4} \approx 0,66$. Finally, the asymptotes are the lines $y = \pm\dfrac{a}{b}x$, that is, $y = \pm\dfrac{4}{3}x$".

Identify and explain in a reasoned manner the two conceptual errors Lucas made, justify why the value obtained for the eccentricity makes no physical or geometric sense based on the definition of a hyperbola, and write the correct solution with the exact values of the foci, eccentricity, and asymptotes.

Determine the standard equation $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ of a hyperbola centered at the origin knowing that its asymptotes are the lines with equations $y = \pm 1,5x$ and that the distance between its two foci is exactly equal to $2\sqrt{13}$ units.

1. a) Find the values of the semi-axes $a$ and $b$, and write the coordinates of its vertices and its foci.
2. b) Calculate the eccentricity of the hyperbola.
3. c) Justify why the eccentricity of any hyperbola depends solely on the slope $m$ of its asymptotes through the relationship $e = \sqrt{1 + m^2}$, and verify that it holds for the obtained data.

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 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 $y = 45\text{ km}$.

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 $2x - 3y = 0$.

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 $P\left(5, \frac{9}{4}\right)$.

A table lamp projects light upwards and downwards 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 centre 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.

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 centre, its foci and its vertices.

Determine the values of the parameter $k$ for which the line $y = x + k$ is tangent, secant or external to the hyperbola $x^2 - 2y^2 = 4$.

Given the hyperbola with standard equation $\dfrac{x^2}{25} - \dfrac{y^2}{144} = 1$:

1. a) Before calculating any point analytically, estimate whether the positive ordinate of the point $P$ on the curve whose abscissa is $x = 13$ will be greater or less than the value $32$. Justify your prediction using exclusively the equation of the asymptote in the first quadrant and the relative position of the hyperbola with respect to it.
2. b) Solve for and calculate the exact value of the positive ordinate of the point $P(13, y)$. Check if your previous estimate was correct and calculate the vertical distance separating the curve from its asymptote at the abscissa $x = 13$.
3. c) Determine the coordinates of the foci $F_1$ and $F_2$ and rigorously verify that for this point $P$, the definition of geometric locus $|d(P, F_1) - d(P, F_2)| = 2a$ holds exactly.

The National Geographic Institute (IGN) has two acoustic recording stations in the Picos de Europa National Park, located at points $A(-5, 0)$ and $B(5, 0)$ (coordinates expressed in kilometers relative to a central point in the valley). During a storm, a rockslide occurs at a point $P(x, y)$ on the terrain. Station $A$ records the roar $20\text{ seconds}$ before station $B$. Assuming that sound travels in a straight line at a constant average speed of $300\text{ m/s}$ ($0,3\text{ km/s}$):

1. a) Model the situation by setting up the geometric locus where the landslide could have originated. Explain why it is a specific branch of a hyperbola and deduce its standard equation, stating what restriction the abscissa $x$ must satisfy.
2. b) An auxiliary seismic station detects that the epicenter of the impact occurred at an ordinate $y = 4\text{ km}$. Determine the exact coordinates of the event's point $P(x, y)$ and calculate the direct straight-line distance from that point to receiving station $A$.

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.

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.

Find the equations of the tangent lines to the hyperbola $x^2 - 4y^2 = 4$ drawn from the exterior point $P(1, 1)$.

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 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 centre 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 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 $B(2\sqrt{3}, 1)$. 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 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.

We want to prove a fundamental geometric property of the hyperbola: "The Euclidean distance from either of the foci of a hyperbola to either of its two asymptotes is always equal to the length of the semi-conjugate axis $b$". For the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$, with focus at $F(c, 0)$ and asymptote $y = \dfrac{b}{a}x$, two students propose different approaches:

• Method 1 (Analytic geometry): Write the asymptote line in its general form $bx - ay = 0$, apply the point-to-line distance formula from the focus $F(c, 0)$, and simplify the expression using the fundamental metric relationship $c^2 = a^2 + b^2$.

• Method 2 (Metric geometry and trigonometry): Draw the perpendicular segment from $F(c, 0)$ to the asymptote, consider the right triangle formed by the origin, the focus, and the foot of the perpendicular, and deduce the distance from the sine of the angle $\alpha$ that the asymptote makes with the positive x-axis.

1. a) Fully develop the proof following both methods.
2. b) Critically compare the two alternatives, analyzing which one is more straightforward algebraically and which one provides clearer visual understanding. Based on the result, describe how to construct with ruler and compass the exact position of the foci if only the vertices $(\pm a, 0)$ and the asymptotes are known.

A hyperbola is called equilateral (or rectangular) when the length of its semi-transverse axis coincides with that of its semi-conjugate axis ($a = b$).

1. a) Formally prove that the eccentricity of any equilateral hyperbola is a universal constant independent of the value of $a$, and determine the angle that its asymptotes form with each other.
2. b) By rotating the Cartesian axes by $45^\circ$, the asymptotes become the coordinate axes themselves and the equation takes the form $xy = k$. Knowing that the vertices $(\pm a, 0)$ of the original hyperbola transform after the rotation into the points $\left(\pm\dfrac{a}{\sqrt{2}}, \pm\dfrac{a}{\sqrt{2}}\right)$, deduce the exact expression of the constant $k$ in terms of the semi-axis $a$.
3. c) Consider the particular equilateral hyperbola $x^2 - y^2 = 32$. Lines passing through the origin of coordinates are drawn, with equation $y = mx$. Reasonably discuss the number of intersection points between the line and the hyperbola as a function of the real parameter $m$, and interpret geometrically what those values of $m$ represent for which no intersection point exists.