Question

4. The distance between a + 7i and -8 + 4i

is Square root of 130 units
The positive value of a is

Answer 1

a = 3

Explanation:

`\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|=√(Re^2+Im^2)\\\\=============================`

`z_1=a+7i,\ z_2=-8+4i\\\\d(z_1,\ z_2)=|(-8+4i)-(a+7i)|=|-8+4i-a-7i|=|-8-a-3i|\\\\=√((-8-a)^2+3^2)\\\\d(z_1,\ z_2)=√(130),\\\\\text{therefore}\ √((-8-a)^2+3^2)=√(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√(121)\\\\8+a=\pm11\qquad\text{subtract 8 from both sides}\\\\a=-19\ or\ a=3`