jmc

Let $w=\frac{z_2}{z_1}.$ Then $w$ is pure imaginary, so $\overline{w}=-w.$

We can write $$\left|\frac{2z_1+7z_2}{2z_1-7z_2}\right|=\left|\frac{2+7\cdot \frac{z_2}{z_1}}{2-7\cdot \frac{z_2}{z_1}}\right|=\left|\frac{2+7w}{2-7w}\right|.$$The conjugate of $2+7w$ is $\overline{2+7w}=2+7\overline{w}=2-7w.$ Thus, the denominator $2-7w$ is the conjugate of the numerator $2+7w,$ which means they have the same absolute value. Therefore, $$\left|\frac{2+7w}{2-7w}\right|=\boxed{1}.$$