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  4. What is the conjugate of (√5+12 i+√5-12i/√5+12i-√5-12i) ?

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Q. What is the conjugate of $\frac{\sqrt{5+12 i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}$ ?

Complex Numbers and Quadratic Equations

Solution:

Let $z=\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}-\sqrt{5-12i}}\times\frac{\sqrt{5+12i}+\sqrt{5-12i}}{\sqrt{5+12i}+\sqrt{5-12i}}$
$=\frac{5+12i+5-12i+2\sqrt{25+144}}{5+12i-5+12i}$
$=\frac{3}{2i}=\frac{3i}{-2}=0-\frac{3}{2}i$
Therefore, the conjugate of $z=0+\frac{3}{2}i$