Question

If the set \(\left( Re \left( \frac{z-\bar{z}+z\bar{z}}{2-3z+5\bar{z}}\right) : z  ∈  C, Re(z)=3 \right)\) is equal to the interval (α,β), then 24(β-α) is equal to

• A. 27
• B. 30
• C. 36
• D. 42

Answer 1

The Correct Option is B

To solve the problem, we need to analyze the expression given in the question step by step.

Step 1: Represent z

Given that Re(z)=3, we can represent z as: z=3+iy where y is the imaginary part of z.

Step 2: Compute z̄

The conjugate of z is: z̄=3−iy

Step 3: Substitute z and z̄ into the expression

We need to compute: Re(z−z̄+zz̄2−3z+5z̄)

First, calculate z−z̄: z−z̄=(3+iy)−(3−iy)=2iy

Next, calculate zz̄: zz̄=(3+iy)(3−iy)=9+y2

Step 4: Calculate the numerator

Now, the numerator becomes: z−z̄+zz̄=2iy+(9+y2)=9+y2+2iy

Step 5: Calculate the denominator

Now, calculate the denominator: 2−3z+5z̄=2−3(3+iy)+5(3−iy)=2−9−3iy+15−5iy=8−8iy

Step 6: Form the complete expression

Now, we can write the complete expression: 9+y2+2iy8−8iy

Step 7: Rationalize the denominator

To find the real part, we multiply the numerator and denominator by the conjugate of the denominator: (9+y2+2iy)(8+8iy)(8−8iy)(8+8iy)

The denominator simplifies to: 64+64y2

The numerator expands to: (9+y2)8+(9+y2)(8iy)+2iy(8)+2iy(8iy)

This results in: 72+8y2+72iy+16iy−16y=72+8y2−16y+(72+16)iy

Step 8: Extract the real part

The real part of the expression is: 72+8y2−16y64+64y2

Step 9: Set the real part to find the interval

To find the interval for t=72+8y2−16y64+64y2, we need to ensure that the expression is non-negative: 72+8y2−16y≥0

This is a quadratic inequality in y.

Step 10: Solve the quadratic inequality

The roots of 8y2−16y+72=0 can be found using the quadratic formula: y=−b±√b2−4ac2a=16±√(−16)2−4⋅8⋅722⋅8

Calculating the discriminant: 256−2304=−2048

Since the discriminant is negative, the quadratic does not cross the x-axis and is always positive.

Step 11: Find the bounds of t

Now, we find the bounds of t: t∈(−18,98)

Step 12: Calculate 24(β−α)

Here, α=−18 and β=98: β−α=98+18=108=54

Thus, 24(β−α)=24⋅54=30

Final Answer

The value of 24(β−α) is 30