Find an equation of the hyperbola having foci at 5Squaruarer

Find an equation of the hyperbola having foci at (5-Squaruareroot 37, 3) and (5+Squareroot 37,3) and asymptotes at y=6x-27 and y = -6x+33

Solution

General equation of hyperbola is (x - h)²/a² - (y - k)²/b² = 1

given focii are (5 - 37 , 3) and (5 + 37 , 3) ; aymptotes are y = 6x - 27 and y = -6x + 33

centre is the midpoint of focii

==> centre (h , k) = (5 , 3)

focii are 37 units either side of centre

==> c = (a² + b²) = 37

==> a² + b² = 37 -------- (1)

slope of asymptotes = +/- (b/a)

==> b/a = 6

==> b = 6a ------ (2)

from (1) and (2)

==> a² + (6a)² = 37

==> 37a² = 37

==> a² = 1

==> a = 1

==> b = 6

equation of hyperbola is (x - 5)²/1² - (y - 3)²/6² = 1

==> (x - 5)²/1 - (y - 3)²/36 = 1