Let α=513+1213i and β=2−3i. The function h(z)=α¯z+β represents a reflection in the complex plane. Find the line that h(z)reflects over.
Give your answer simplified in the form pa+qb=r where a=Re(z) and b=Im(z).
Thanks
1. Understand the Reflection
General Form: A reflection in the complex plane can be represented by the function: h(z) = α * conj(z) + β where: * z is the complex number being reflected
* conj(z) is the complex conjugate of z * α is a complex number that determines the scaling and rotation * β is a complex number that determines the translation
Key Insight: The line of reflection is perpendicular to the vector represented by α.
2. Determine the Vector α
α = 5/13 + 12/13 * i
3. Find the Line of Reflection
Perpendicular Vector: The vector perpendicular to α is found by rotating α by 90 degrees counterclockwise.
We can achieve this by multiplying α by i: i * α = i * (5/13 + 12/13 * i) = -12/13 + 5/13 * i
Line Equation: The line of reflection passes through the origin and is parallel to the vector -12/13 + 5/13 * i.
Equation: y = (5/12) * x or 5x - 12y = 0
Therefore, the line of reflection for the function h(z) = α*conj(z) + β is 5x - 12y = 0.
Note: The β term (2 - 3i) in the function h(z) translates the reflection, but it does not change the line of reflection itself. The line of reflection remains the same regardless of the translation.
1. Understand the Reflection
General Form: A reflection in the complex plane can be represented by the function: h(z) = α * conj(z) + β where: * z is the complex number being reflected
* conj(z) is the complex conjugate of z * α is a complex number that determines the scaling and rotation * β is a complex number that determines the translation
Key Insight: The line of reflection is perpendicular to the vector represented by α.
2. Determine the Vector α
α = 5/13 + 12/13 * i
3. Find the Line of Reflection
Perpendicular Vector: The vector perpendicular to α is found by rotating α by 90 degrees counterclockwise.
We can achieve this by multiplying α by i: i * α = i * (5/13 + 12/13 * i) = -12/13 + 5/13 * i
Line Equation: The line of reflection passes through the origin and is parallel to the vector -12/13 + 5/13 * i.
Equation: y = (5/12) * x or 5x - 12y = 0
Therefore, the line of reflection for the function h(z) = α*conj(z) + β is 5x - 12y = 0.
Note: The β term (2 - 3i) in the function h(z) translates the reflection, but it does not change the line of reflection itself. The line of reflection remains the same regardless of the translation.