The lengths of the tangents from $\mathrm{A}(-3,5)$ and B(1, 2) to a circle are 6 and 5 respectively. Then the length of the tangent to the circle drawn from the point $\left(5,-1\right)$ is

$\mathrm{Let} \mathrm{S}={\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0 \mathrm{be} \mathrm{the} \mathrm{equation} \mathrm{of} \mathrm{circle}$

Length of tangent  from $\mathrm{A}(-3,5)$ is 6

$$\begin{gathered} 9+25-6\mathrm{g}+10\mathrm{f}+\mathrm{c}=36 \\ \Rightarrow -6\mathrm{g}+10\mathrm{f}+\mathrm{c}=2\dots .\left(1\right) \end{gathered}$$

Length of tangent  from  B(1, 2) to  circle is 5

$\begin{array}{l}1+4+2\mathrm{g}+4\mathrm{f}+\mathrm{c}=25\\ \Rightarrow 2\mathrm{g}+4\mathrm{f}+\mathrm{c}=20\dots \dots \dots \left(2\right)\end{array}$

$\begin{array}{l}2\times \left(2\right)-\left(1\right)\Rightarrow 10\mathrm{g}-2\mathrm{f}+\mathrm{c}=38\\ \Rightarrow 26+10\mathrm{g}-2\mathrm{f}+\mathrm{c}=64\end{array}$

The length of the tangent to the circle drawn from the point $\left(5,-1\right)$ is 8