Question

Equation of hyperbola passing through the point (1, -1) and having asymptotes x + 2y + 3 = 0 and 3x + 4y + 5 = 0 is

Answer 1

$\textbf{Given:}$

$\text{Asymptotes are x+2y+3=0 and 3x+4y+5=0}$

$\text{Combined equation of asymptotes is}(x+2y+3)(3x+4y+5)=0$

$\textbf{We know that the combined equation of asymptotes and the equation}$ $\textbf{of hyberbola differs only by constant term.}$

$\text{Equation of hyberola is}(x+2y+3)(3x+4y+5)+k=0$

$\text{It passes though(1,-1) }$

$(1+2(-1)+3)(3+4(-1)+5)+k=0$

$\implies(2)(4)+k=0$

$\implies\boxed{\bf{k=-8}}$

$\therefore\text{Equation of required hyberbola is }\bf\,(x+2y+3)(3x+4y+5)-8=0$