If the angle between the circles $x^2+y^2-4x-6y-3=0,x^2+y^2+8x-4y+\lambda =0$ is ${60}^0$ then the value of $\lambda$ is

$$\begin{gathered} \mathrm{cos\theta }=\frac{\mathrm{c}+{\mathrm{c}}^{\text{'}}-2{\mathrm{gg}}^{\text{'}}-2{\mathrm{ff}}^{\text{'}}}{2\sqrt{{\mathrm{g}}^2+{\mathrm{f}}^2-\mathrm{c}} \sqrt{{(\mathrm{g}\text{'})}^2+{(\mathrm{f}\text{'})}^2-{\mathrm{c}}^{\text{'}}}} \\ \frac{1}{2}=\frac{-3+\mathrm{\lambda }+4\left(4\right)-12}{2\sqrt{16} \sqrt{16+4-\mathrm{\lambda }}} \\ 1=\frac{\mathrm{\lambda }+1}{4\sqrt{20-\mathrm{\lambda }}} \\ 4\left(\sqrt{20-\mathrm{\lambda }}\right)=\mathrm{\lambda }+1 \\ \mathrm{squating} \mathrm{on} b\mathrm{oth} \mathrm{side}s \\ 16(20-\mathrm{\lambda })={\mathrm{\lambda }}^2+1+2\mathrm{\lambda } \\ {\mathrm{\lambda }}^2+18\mathrm{\lambda }-319=0 \\ \mathrm{simplify} \mathrm{we} \mathrm{get} \mathrm{\lambda }=-29 \end{gathered}$$