COMEDK · Maths · 3. Complex Number
If \(1+i=(x+i y)(u+i v)\), then \(\tan ^{-1}\left(\frac{y}{x}\right)+\cot ^{-1}\left(\frac{u}{v}\right)\) has the value
- A \(n \pi+\frac{\pi}{6}, n \in I\)
- B \(2 n \pi+\frac{\pi}{3}, n \in I\)
- C \(n \pi+\frac{\pi}{4}, n \in I\)
- D \(n \pi-\frac{\pi}{3}, n \in I\)
Answer & Solution
Correct Answer
(C) \(n \pi+\frac{\pi}{4}, n \in I\)
Step-by-step Solution
Detailed explanation
Given, equation is \[ \begin{gathered} 1+i=(x+i y)(u+i v) \\ \Rightarrow 1+i=(x u-y v)+i(x v+y u) \end{gathered} \] Comparing real and imaginary parts, we get \[ \begin{aligned} &x u-y v=1 \quad \text{...(i)}\\ &x v+y u=1 \quad \text{...(ii)} \end{aligned} \] Multiply Eq. (i) by…
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