What is the value of k if the midpoint between the points(a + b, a - b) and (- a, b) lies on the line ax + by = k?

We know, Mid-point of points (x₁, y₁) &  (x₂, y₂) is $\left(\frac{{\mathrm{x}}_1+{\mathrm{x}}_2}{2},\frac{{\mathrm{y}}_1+{\mathrm{y}}_2}{2}\right)$.

Hence, the mid-point of (a + b, a - b) and (- a, b) is $\left(\frac{\mathrm{a}+\mathrm{b}-\mathrm{a}}{2},\frac{\mathrm{a}-\mathrm{b}+\mathrm{b}}{2}\right)$ $=\left(\frac{\mathrm{b}}{2},\frac{\mathrm{a}}{2}\right)$

Given: Mid-point lies on the line ax + by = k

$\Rightarrow \mathrm{a}\times \frac{\mathrm{b}}{2}+\mathrm{b}\times \frac{\mathrm{a}}{2}=\mathrm{k}$

$\Rightarrow \frac{\mathrm{ab}}{2}+\frac{\mathrm{ab}}{2}=\mathrm{k}$

$\Rightarrow \mathrm{ab}=\mathrm{k}$

Thus, the value of k = ab