Complex Numbers
Rotation – Reflection via Complex Conjugate
Complex Numbers_PYQ
Grade 11
Question:
If $0 < \alpha < \dfrac{\pi}{2}$ is a fixed angle. If $P=(\cos\theta,\sin\theta)$ and $Q=\{\cos(\alpha-\theta),\sin(\alpha-\theta)\}$, then $Q$ is obtained from $P$ by
clockwise rotation around origin through an angle $\alpha$
anti-clockwise rotation around origin through an angle $\alpha$
reflection in the line through origin with slope $\tan\alpha$
reflection in the line through origin with slope $\tan\dfrac{\alpha}{2}$
Step-by-Step Solution
Key Concept: Reflection of $z$ in the line through origin at angle $\phi$ is $e^{2i\phi}\bar{z}$. Since $Q=e^{i\alpha}\bar{P}$, we need $2\phi=\alpha$, i.e., $\phi=\alpha/2$, giving the line with slope $\tan(\alpha/2)$.
**Step 1: Write P and Q as complex numbers**
$P = e^{i\theta}$, $Q = e^{i(\alpha-\theta)} = e^{i\alpha}\cdot e^{-i\theta} = e^{i\alpha}\cdot\bar{P}$.
**Step 2: Identify the transformation**
The reflection of $z$ in the line through the origin at angle $\phi$ is $z_{\text{refl}} = e^{2i\phi}\,\bar{z}$. Setting $\phi=\alpha/2$: $z_{\text{refl}}=e^{i\alpha}\bar{P}=Q$. ✓
**Step 3: Identify the line**
The line at angle $\alpha/2$ has slope $\tan(\alpha/2)$.
Correct Answer: 4