2x + y = 0 is the equation of a diameter of the circle which touches the lines 4x – 3y + 10 = 0 and 4x – 3y – 30 = 0. The centre and radius of the circle are

$$\begin{gathered} diameter=2r=\frac{\left|10+30\right|}{\sqrt{16+9}}=8\Rightarrow r=4 \\ \mathrm{Equation} \mathrm{of} \mathrm{a} \mathrm{line} \mathrm{midway} \mathrm{between} 4\mathrm{x} – 3\mathrm{y} + 10 = 0 \mathrm{and} 4\mathrm{x} – 3\mathrm{y} – 30 = 0 \\ \mathrm{is}4\mathrm{x}-3\mathrm{y}+\frac{10-30}{2}=0(\mathrm{i}.\mathrm{e})4\mathrm{x}-3\mathrm{y}-10=0 \\ \therefore \mathrm{centre} \mathrm{is} \mathrm{point} \mathrm{of} \mathrm{intersection} \mathrm{of} 2\mathrm{x} + \mathrm{y} = 0 \mathrm{and} 4\mathrm{x}-3\mathrm{y}-10=0 \\ \mathrm{centre}=(1,-2) \end{gathered}$$