The standard form of a hyperbola is
color(white)("XXX")(x-h)^2/(a^2)-(y-k)^2/(b^2) = 1
Given
color(white)("XXX")4x^2-5y-16x-30y-9=0
Group the x terms and the y terms separately and transfer the constant to the right side of the equation.
color(white)("XXX")color(red)(4x^2-16x)-color(blue)(5y^2+30y) = color(green)(9)
Extract the constant factors from each of the x and y sub-expressions.
color(white)("XXX")color(red)(4(x^2-4x))-color(blue)(5(y^2+6y)) = color(green)(9)
Complete the squares for each of the x and y sub-expressions
color(white)("XXX")color(red)(4(x^2-4x+2^2))-color(blue)(5(y^2+6y+3^2)) = color(green)(9)+color(red)(16)-color(blue)(45)
color(white)("XXX")color(red)(4(x-2)^2)-color(blue)(5(y+3)^2) = -20
Dividing both sides by (-20)
color(white)("XXX")color(red)(-(x-2)^2/5)color(blue)(+(y+3)^2/4) = 1
Convert the denominators into squares and re-order the left side terms to avoid the ugly leading minus sign.
color(white)("XXX")color(blue)((y+3)^2/2^2)-color(red)((x-2)^2/(sqrt(5))^2) = 1
Note that the reversal of the x and y terms from the normal form implies that the transverse axis is vertical.
graph{4x^2-5y^2-16x-30y-9=0 [-9.14, 19.34, -9.06, 5.18]}
