Question

The value of \(\frac{5+5i}{3-4i} + \frac{20}{4+3i}\) is given by :

• A. \(3 - i\)
• B. \(1 + i\)
• C. \(\frac{5}{2} - \frac{i}{2}\)
• D. \(-2 + 9i\)

Hint:

To perform this faster:
Note that both denominators (\(3-4i\) and \(4+3i\)) have a modulus squared of 25.
We can write a common denominator of 25:
\[ \frac{5(1+i)(3+4i) + 20(4-3i)}{25} \]
\[ = \frac{5(-1+7i) + 80 - 60i}{25} = \frac{-5 + 35i + 80 - 60i}{25} = \frac{75 - 25i}{25} = 3 - i \]

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
To evaluate an expression involving complex numbers in the denominator, we must rationalize each term.
This is done by multiplying both the numerator and the denominator of the fraction by the complex conjugate of the denominator.
Key Formula or Approach:
For any complex fraction \(\frac{a+ib}{c+id}\), we multiply by the conjugate \((c-id)\):
\[ \frac{a+ib}{c+id} = \frac{(a+ib)(c-id)}{(c+id)(c-id)} = \frac{(ac+bd) + i(bc-ad)}{c^2 + d^2} \]

Step 2: Detailed Explanation:
Let us calculate each of the two terms separately:
- Term 1: \(\frac{5+5i}{3-4i}\)
Multiply numerator and denominator by the conjugate of the denominator, which is \(3+4i\):
\[ \frac{5+5i}{3-4i} = \frac{(5+5i)(3+4i)}{(3-4i)(3+4i)} \]
Expand the numerator:
\[ (5+5i)(3+4i) = 15 + 20i + 15i + 20i^2 = 15 + 35i - 20 = -5 + 35i \]
Calculate the denominator:
\[ 3^2 + (-4)^2 = 9 + 16 = 25 \]
So, Term 1 is:
\[ \frac{-5 + 35i}{25} = -\frac{5}{25} + \frac{35}{25}i = -\frac{1}{5} + \frac{7}{5}i \]
- Term 2: \(\frac{20}{4+3i}\)
Multiply numerator and denominator by the conjugate of the denominator, which is \(4-3i\):
\[ \frac{20}{4+3i} = \frac{20(4-3i)}{(4+3i)(4-3i)} \]
Expand the numerator:
\[ 20(4-3i) = 80 - 60i \]
Calculate the denominator:
\[ 4^2 + 3^2 = 16 + 9 = 25 \]
So, Term 2 is:
\[ \frac{80 - 60i}{25} = \frac{80}{25} - \frac{60}{25}i = \frac{16}{5} - \frac{12}{5}i \]
Now, sum the two terms:
\[ \text{Total} = \left(-\frac{1}{5} + \frac{7}{5}i\right) + \left(\frac{16}{5} - \frac{12}{5}i\right) \]
Group real and imaginary parts:
\[ \text{Real Part} = -\frac{1}{5} + \frac{16}{5} = \frac{15}{5} = 3 \]
\[ \text{Imaginary Part} = \frac{7}{5}i - \frac{12}{5}i = -\frac{5}{5}i = -i \]
Thus, the sum is \(3 - i\). This matches Option (A).

Step 3: Final Answer:
The correct option is (A).