Given Quotient:-
[tex]\red{➤}\:[/tex][tex]\sf \dfrac{13}{5-i} [/tex]
[tex]\\[/tex]
To Do:-
[tex]\orange{☛}\:[/tex][tex]\sf Write \: given \: Quotient \:in \: a+bi \:form[/tex]
[tex]\\[/tex]
Solution:-
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{13}{5-i} \\\end{gathered} [/tex]
Multiplaying denominator and numerator by 5+i to get denominator as real number
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{13(5+i)}{5-i(5+i)} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{5^2-i^2} \\\end{gathered} [/tex]
[tex]\quad\quad\quad\quad\quad\sf ((a+b)(a-b)=a^2-b^2) [/tex]
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{25-(-1)} \\\end{gathered} [/tex]
[tex]\quad\quad\quad\quad\quad\sf (For\:Complex\: Number,i ; i^2 = -1) [/tex]
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{25+1} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{65+13i}{26} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\quad\longrightarrow\quad\sf \frac{\cancel{65}}{\cancel{26}}+\frac{\cancel{13}i}{\cancel{26}} \\\end{gathered} [/tex]
[tex]\begin{gathered}\\\quad\longrightarrow\quad\boxed{\sf {\frac{5}{2} +\frac{1}{2} i}}\\\end{gathered} [/tex]
