If the angle between the circles $x^2+y^2+4x-5=0$ and $x^2+y^2+2\lambda y-4=0$ is $\frac{\pi }{3}$ then $\lambda =$

$\text{The equation of the circles is given as, }$

$x^2+y^2+4x-5=0$

$x^2+y^2+2\lambda y-4=0$

$\text{On comparing in equations with }$$x^2+y^2+2gx+2fy+c=0$

$g_1=2,f_1=0,c_1=-5$

$g_2=0,f_2=\lambda ,c_2=-4$

$\text{The angle between the circles is, }$

$\cos \theta =\frac{2g_1{g}_2+2f_1{f}_2-c_1-c_2}{2\sqrt{g_1^2+f_1^2-c_1}\sqrt{g_2^2+f_2^2-c_2}}$

$\cos \frac{\pi }{3}=\frac{2\times 2\times 0+2\times 0\times \lambda +5+4}{2\sqrt{4+0+5}\sqrt{0+{\lambda }^2+4}}$

$\frac{1}{2}=\frac{3}{2\sqrt{{\lambda }^2+4}}$

$\text{After simplification, }$

$\lambda =\pm \sqrt{5}$