For the hyperbola \(\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((\pm 5, 0)\)
Vertices \((\pm 4, 0)\)
\(e = \dfrac{5}{4}\)
Latus rectum = \(\dfrac{9}{2}\)
For the hyperbola \(\dfrac{y^2}{9} - \dfrac{x^2}{27} = 1\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((0, \pm 6)\)
Vertices \((0, \pm 3)\)
\(e = 2\)
Latus rectum = 18
For the hyperbola \(9y^2 - 4x^2 = 36\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((0, \pm \sqrt{13})\)
Vertices \((0, \pm 2)\)
\(e = \dfrac{\sqrt{13}}{2}\)
Latus rectum = 9
For the hyperbola \(16x^2 - 9y^2 = 576\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((\pm 10, 0)\)
Vertices \((\pm 6, 0)\)
\(e = \dfrac{5}{3}\)
Latus rectum = \(\dfrac{64}{3}\)
For the hyperbola \(5y^2 - 9x^2 = 36\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \(\left(0, \pm \dfrac{2\sqrt{14}}{\sqrt{5}}\right)\)
Vertices \(\left(0, \pm \dfrac{6}{\sqrt{5}}\right)\)
\(e = \dfrac{\sqrt{14}}{3}\)
Latus rectum = \(\dfrac{4\sqrt{5}}{3}\)
For the hyperbola \(49y^2 - 16x^2 = 784\), find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum.
Foci \((0, \pm \sqrt{65})\)
Vertices \((0, \pm 4)\)
\(e = \dfrac{\sqrt{65}}{4}\)
Latus rectum = \(\dfrac{49}{2}\)
Find the equation of the hyperbola with vertices \((\pm 2, 0)\) and foci \((\pm 3, 0)\).
\(\dfrac{x^2}{4} - \dfrac{y^2}{5} = 1\)
Find the equation of the hyperbola with vertices \((0, \pm 5)\) and foci \((0, \pm 8)\).
\(\dfrac{y^2}{25} - \dfrac{x^2}{39} = 1\)
Find the equation of the hyperbola with vertices \((0, \pm 3)\) and foci \((0, \pm 5)\).
\(\dfrac{y^2}{9} - \dfrac{x^2}{16} = 1\)
Find the equation of the hyperbola whose foci are \((\pm 5, 0)\) and whose transverse axis is of length 8.
\(\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1\)
Find the equation of the hyperbola whose foci are \((0, \pm 13)\) and whose conjugate axis is of length 24.
\(\dfrac{y^2}{25} - \dfrac{x^2}{144} = 1\)
Find the equation of the hyperbola whose foci are \((\pm 3\sqrt{5}, 0)\) and whose latus rectum is of length 8.
\(\dfrac{x^2}{25} - \dfrac{y^2}{20} = 1\)
Find the equation of the hyperbola whose foci are \((\pm 4, 0)\) and whose latus rectum is of length 12.
\(\dfrac{x^2}{4} - \dfrac{y^2}{12} = 1\)
Find the equation of the hyperbola whose vertices are \((\pm 7, 0)\) and whose eccentricity is \(e = \dfrac{4}{3}\).
\(\dfrac{x^2}{49} - \dfrac{9y^2}{343} = 1\)
Find the equation of the hyperbola whose foci are \((0, \pm \sqrt{10})\) and which passes through the point \((2, 3)\).
\(\dfrac{y^2}{5} - \dfrac{x^2}{5} = 1\)