Answer:

The value of k =  [tex]\frac{9}{4}[/tex]

Step-by-step explanation:

The equation of the hyperbola is

4x² - y² = 36

[tex]\frac{4x^{2} }{36} -\frac{y^{2} }{36} =\frac{36}{36}\\\frac{4x^{2} }{36} -\frac{y^{2} }{36} =1[/tex]

[tex]\frac{x^{2} }{9} -\frac{y^{2} }{36}=1[/tex] --------------(1)

a² = 9

b² = 36

The equation of the line is

5x -2y + 4k = 0

2y = 5x + 4k

y = ([tex]\frac{5}{2}[/tex])x + 4k  ------------------(2)

From equation(2)

m =  ([tex]\frac{5}{2}[/tex]) and c= 2k

c² = a²m² - b²

(2k)² = 9 ([tex]\frac{5}{2}[/tex])² - 36

4k² = [tex]\frac{225}{4}[/tex] - 36

4k² = [tex]\frac{225-144}{4}[/tex]

k² = [tex]\frac{81}{16}[/tex]

k = [tex]\frac{9}{4}[/tex]

Therefore, k = [tex]\frac{9}{4}[/tex]