A line moves such that the portion of it intercepted between the coordinate axes is of constant length a, then the locus of the mid point of that line segment is

Let the line be $\frac{\mathrm{x}}{\mathrm{p}}+\frac{\mathrm{y}}{8}=1$ which cuts coordinate axes at A=(P, O), B=(o, q) respectively.

Given that AB=a

$$\begin{gathered} \Rightarrow \sqrt{{\mathrm{p}}^2+{\mathrm{q}}^2}=\mathrm{a} \\ \Rightarrow {\mathrm{p}}^2+{\mathrm{q}}^2={\mathrm{a}}^2 - \left(1\right) \\ \mathrm{let} \mathrm{P}=(\mathrm{x}, \mathrm{y}) \mathrm{be} \mathrm{locus} \mathrm{of} \mathrm{mid} \mathrm{point} \mathrm{of} \overline{\mathrm{AB}} \\ \Rightarrow (\mathrm{x}, \mathrm{y})=\left(\frac{\mathrm{p}}{2}, \frac{\mathrm{q}}{2}\right) \\ \Rightarrow \mathrm{P}=2\mathrm{x}, \mathrm{q}=2\mathrm{y} \\ \mathrm{from} \left(1\right) \Rightarrow 4{\mathrm{x}}^2+4{\mathrm{y}}^2={\mathrm{a}}^2 \\ \Rightarrow {\mathrm{x}}^2+{\mathrm{y}}^2=\frac{{\mathrm{a}}^2}{4} \end{gathered}$$