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The area of the triangle formed by the complex number z, iz, z + iz in the argand plane is
- a)(1/2) |z|2
- b)|z|2
- c)2 |z|2
- d)none
Correct answer is option 'A'. Can you explain this answer?
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The area of the triangle formed by the complex number z, iz, z + iz in...
Area, AABC =
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The area of the triangle formed by the complex number z, iz, z + iz in...
The area of the triangle formed by the complex numbers z, iz, and z + iz in the Argand plane can be found using the Shoelace formula.
Shoelace Formula:
The Shoelace formula is a method used to calculate the area of a polygon given the coordinates of its vertices in the Cartesian plane. The formula states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) is given by:
Area = 1/2 * |(x1y2 + x2y3 + ... + xn-1yn + xn*y1) - (y1x2 + y2x3 + ... + yn-1xn + yn*x1)|
Applying the Shoelace Formula:
Let's apply the Shoelace formula to the triangle formed by the complex numbers z, iz, and z + iz.
We have the vertices of the triangle as:
A = z
B = iz
C = z + iz
The coordinates of these complex numbers in the Argand plane are:
A = (Re(z), Im(z))
B = (Re(iz), Im(iz))
C = (Re(z + iz), Im(z + iz))
Calculating the Area:
Using the Shoelace formula, the area of the triangle is:
Area = 1/2 * |(Re(z) * Im(iz) + Re(iz) * Im(z + iz) + Re(z + iz) * Im(z)) - (Im(z) * Re(iz) + Im(iz) * Re(z + iz) + Im(z + iz) * Re(z))|
Simplifying this expression further, we get:
Area = 1/2 * |(Re(z) * Im(iz) + Re(iz) * (Im(z) + Im(iz)) + (Re(z) + Re(iz)) * Im(z)) - (Im(z) * Re(iz) + Im(iz) * (Re(z) + Re(iz)) + (Im(z) + Im(iz)) * Re(z))|
Area = 1/2 * |(Re(z) * Im(iz) + Re(iz) * Im(z) + Re(iz) * Im(iz) + Re(z) * Im(z) + Re(z) * Im(iz)) - (Im(z) * Re(iz) + Im(iz) * Re(z) + Im(iz) * Re(iz) + Im(z) * Re(z) + Im(z) * Re(iz))|
Area = 1/2 * |(2 * Re(z) * Im(iz) + Re(iz) * Im(iz) + Re(z) * Im(iz)) - (2 * Im(z) * Re(iz) + Im(iz) * Re(iz) + Im(z) * Re(iz))|
Using the properties of complex numbers, we can simplify further:
Re(iz) = -Im(z)
Im(iz) = Re(z)
Substituting these values, we get:
Area = 1/2 * |(2 * Re(z) * Re(z) + Re(z) * Re(z) + Re(z) * Re(z)) - (2
Shoelace Formula:
The Shoelace formula is a method used to calculate the area of a polygon given the coordinates of its vertices in the Cartesian plane. The formula states that the area of a polygon with vertices (x1, y1), (x2, y2), ..., (xn, yn) is given by:
Area = 1/2 * |(x1y2 + x2y3 + ... + xn-1yn + xn*y1) - (y1x2 + y2x3 + ... + yn-1xn + yn*x1)|
Applying the Shoelace Formula:
Let's apply the Shoelace formula to the triangle formed by the complex numbers z, iz, and z + iz.
We have the vertices of the triangle as:
A = z
B = iz
C = z + iz
The coordinates of these complex numbers in the Argand plane are:
A = (Re(z), Im(z))
B = (Re(iz), Im(iz))
C = (Re(z + iz), Im(z + iz))
Calculating the Area:
Using the Shoelace formula, the area of the triangle is:
Area = 1/2 * |(Re(z) * Im(iz) + Re(iz) * Im(z + iz) + Re(z + iz) * Im(z)) - (Im(z) * Re(iz) + Im(iz) * Re(z + iz) + Im(z + iz) * Re(z))|
Simplifying this expression further, we get:
Area = 1/2 * |(Re(z) * Im(iz) + Re(iz) * (Im(z) + Im(iz)) + (Re(z) + Re(iz)) * Im(z)) - (Im(z) * Re(iz) + Im(iz) * (Re(z) + Re(iz)) + (Im(z) + Im(iz)) * Re(z))|
Area = 1/2 * |(Re(z) * Im(iz) + Re(iz) * Im(z) + Re(iz) * Im(iz) + Re(z) * Im(z) + Re(z) * Im(iz)) - (Im(z) * Re(iz) + Im(iz) * Re(z) + Im(iz) * Re(iz) + Im(z) * Re(z) + Im(z) * Re(iz))|
Area = 1/2 * |(2 * Re(z) * Im(iz) + Re(iz) * Im(iz) + Re(z) * Im(iz)) - (2 * Im(z) * Re(iz) + Im(iz) * Re(iz) + Im(z) * Re(iz))|
Using the properties of complex numbers, we can simplify further:
Re(iz) = -Im(z)
Im(iz) = Re(z)
Substituting these values, we get:
Area = 1/2 * |(2 * Re(z) * Re(z) + Re(z) * Re(z) + Re(z) * Re(z)) - (2
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