Question

Find the coordinates of the foci and vertices, the endpoints of the transverse axis, conjugate axis, and latus rectum for each hyperbola, whose center is at (0,0). Draw the hyperbola and its directrices. Show your solution. (x²)/(25)-(y²))/(4)=1

Answer 1

The given hyperbola's foci are at approximately (±5.385, 0), vertices and endpoints of transverse axis at (±5, 0), and endpoints of conjugate axis at (0, ±2). The length of the latus rectum is 8/5, and the directrices are vertical lines at x ≈ ±4.618.

Step-by-step explanation:

The equation of the hyperbola (x²/25) - (y²/4) = 1 indicates that its center is at (0,0), with a² = 25 and b² = 4. The vertices are at (±5,0), and the foci are located at (±c,0) where c² = a² + b² = 29, hence c ≈ ±5.385. The endpoints of the transverse axis are the vertices, and those of the conjugate axis are at (0, ±2). Length of the latus rectum is 2b²/a = 8/5. The hyperbola opens sideways since a² is under x². The directrices are vertical lines at x = ±a/e ≈ ±4.618, where e = c/a.