Question

Question 4 Find the coordinates of the foci and the vertices, the eccentricity, and the length of the latus rectum of the hyperbola 16x^2 – 9y^2 = 576

Class X1 - Maths -Conic Sections Page 262

Answer 1

16x² - 9y² = 576
dividing by 576 from both sides,
16x²/576 - 9y²/576 = 576/576
x²/{576/16} - y²/{576/9} = 1
x²/36 - y²/64 = 1
x²/6² - y²/8² = 1 ---------------(1)

concept : if equation of hyperbola is in the form of x²/a² - y²/b² = 1-----------(2) then,
vertices ( ±a, 0)
foci ( ±c , 0) where, c² = a² + b²
eccentricity ( e ) = c/a
latusrectum = 2b²/a

now, compare equations (1) and (2),
a = 6 and b = 8
then , c = √(a² + b²) = √(6² + 8²) = 10

hence,
vertices ( ±a , 0) = ( ± 6, 0)
Foci ( ± c , 0) = ( ±10, 0)
eccentricity ( e ) = c/a = 10/6 = 5/3
Latusrectum = 2b²/a = 2 × 64/6 = 64/3