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:

Notice that the denominators \(3-4i\) and \(4+3i\) have the same modulus squared: \(3^2+4^2 = 25\).
We can write \((4+3i) = i(3-4i)\).
So: \[ \frac{20}{4+3i} = \frac{20}{i(3-4i)} = \frac{-20i}{3-4i} \] Adding the two terms with the same denominator:
\[ \frac{5+5i - 20i}{3-4i} = \frac{5-15i}{3-4i} = \frac{5(1-3i)(3+4i)}{25} = \frac{5(3+4i-9i+12)}{25} = \frac{5(15-5i)}{25} = 3-i \]

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to simplify the sum of two complex fractions: \(\frac{5+5i}{3-4i}\) and \(\frac{20}{4+3i}\).
We can do this by rationalizing each denominator (multiplying by the complex conjugate).
Key Formula or Approach:
To simplify \(\frac{a+bi}{c+di}\), multiply the numerator and denominator by the conjugate \(c-di\):
\[ \frac{a+bi}{c+di} = \frac{(a+bi)(c-di)}{c^2 + d^2} \]

Step 2: Detailed Explanation:

• Simplify the first term \(\frac{5+5i}{3-4i}\):
Multiply the numerator and denominator by the conjugate \(3+4i\):
\[ \frac{5+5i}{3-4i} = \frac{(5+5i)(3+4i)}{(3-4i)(3+4i)} \] Calculate 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, the first term becomes:
\[ \frac{-5 + 35i}{25} = -\frac{5}{25} + \frac{35}{25}i = -\frac{1}{5} + \frac{7}{5}i \]

• Simplify the second term \(\frac{20}{4+3i}\):
Multiply the numerator and denominator by the conjugate \(4-3i\):
\[ \frac{20}{4+3i} = \frac{20(4-3i)}{(4+3i)(4-3i)} \] Calculate the numerator:
\[ 20(4-3i) = 80 - 60i \] Calculate the denominator:
\[ 4^2 + 3^2 = 16 + 9 = 25 \] So, the second term becomes:
\[ \frac{80 - 60i}{25} = \frac{80}{25} - \frac{60}{25}i = \frac{16}{5} - \frac{12}{5}i \]

• Now, add the two simplified terms:
\[ \text{Sum} = \left(-\frac{1}{5} + \frac{7}{5}i\right) + \left(\frac{16}{5} - \frac{12}{5}i\right) \] Combine the real parts and the imaginary parts: \[ \text{Real part} = -\frac{1}{5} + \frac{16}{5} = \frac{15}{5} = 3 \] \[ \text{Imaginary part} = \frac{7}{5}i - \frac{12}{5}i = -\frac{5}{5}i = -i \] \[ \text{Sum} = 3 - i \]

Step 3: Final Answer:
The value of the expression is \(3-i\).