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 $C(6, -3)$ are interior, exterior, or if they belong to the hyperbola with equation $\frac{x^2}{16} - \frac{y^2}{9} = 1$.

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.

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 that passes through the point $P(5, 4)$.

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.

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

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 $B$ 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 $y = 45\text{ mi}$.

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 $2x - 3y = 0$.

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.

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

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.

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.

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

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.

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.

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

Prove analytically that the area of the triangle formed by any tangent line to the rectangular hyperbola with equation $xy = k$ (where $k > 0$) and the two coordinate axes is independent of the point of tangency. Calculate this constant value in terms of the parameter $k$.

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.

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 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.

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.