Answer:
a = 3
Step-by-step explanation:
[tex]\text{The formula of a distance between two complex numbers}\\\\z_1=a+bi,\ z_2=c+di:\\\\d(z_1,\ z_2)=|z_2-z_1|\ \text{where}\ |z|=\sqrt{Re^2+Im^2}\\\\=============================[/tex]
[tex]z_1=a+7i,\ z_2=-8+4i\\\\d(z_1,\ z_2)=|(-8+4i)-(a+7i)|=|-8+4i-a-7i|=|-8-a-3i|\\\\=\sqrt{(-8-a)^2+3^2}\\\\d(z_1,\ z_2)=\sqrt{130},\\\\\text{therefore}\ \sqrt{(-8-a)^2+3^2}=\sqrt{130}\Rightarrow(-8-a)^2+3^2=130\\\\\bigg(-(8+a)\bigg)^2+9=130\qquad\text{subtract 9 from both sides}\\\\(8+a)^2=121\iff8+a=\pm\sqrt{121}\\\\8+a=\pm11\qquad\text{subtract 8 from both sides}\\\\a=-19\ or\ a=3[/tex]
