Given z = (a+jb)/(c+jd), where a, b, c, d are real quantities, show that if z is real, then (a)/(b) = (c)/(d), and if z is entirely imaginary, then ac = -bd.

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Asked on May 17, 2026Mathematics

This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.

Accepted Answer · 2026-05-17T14:39:54.637Z

Here are the solutions for the remaining questions. Question 11: Given z = (a+jb)/(c+jd), where a, b, c, d are real quantities. Step 1: Express z in the form X + jY by multiplying the numerator and denominator by the conjugate of the denominator. z = (a+jb)/(c+jd) × (c-jd)/(c-jd) z = ((a+jb)(c-jd))/(c^2 - (jd)^2) z = (ac - ajd + jbc - j^2bd)/(c^2 + d^2) Since j^2 = -1: z = (ac - ajd + jbc + bd)/(c^2 + d^2) Group the real and imaginary terms: z = ((ac + bd) + j(bc - ad))/(c^2 + d^2) z = (ac + bd)/(c^2 + d^2) + j(bc - ad)/(c^2 + d^2) a) If z is real, its imaginary part must be zero. Step 2: Set the imaginary part of z to zero. (bc - ad)/(c^2 + d^2) = 0 Since c^2 + d^2 ≠ 0, the numerator must be zero: bc - ad = 0 bc = ad Step 3: Rearrange the equation to show the required ratio. Assuming b ≠ 0 and d ≠ 0, divide both sides by bd: (bc)/(bd) = (ad)/(bd) (c)/(d) = (a)/(b) Thus, if z is real, then (a)/(b) = (c)/(d). b) If z is entirely imaginary, its real part must be zero. Step 4: Set the real part of z to zero. (ac + bd)/(c^2 + d^2) = 0 Since c^2 + d^2 ≠ 0, the numerator must be zero: ac + bd = 0 ac = -bd Step 5: Rearrange the equation to show the required ratio. Assuming b ≠ 0 and c ≠ 0, divide both sides by bc: (ac)/(bc) = -(bd)/(bc) (a)/(b) = -(d)/(c) Thus, if z is entirely imaginary, then (a)/(b) = -(d)/(c). Question 12: Given (a+b) + j(a-b) = (1+j)^2 + j(2+j), obtain the values of a and b. Step 1: Simplify the right-hand side of the equation. First, expand (1+j)^2: (1+j)^2 = 1^2 + 2(1)(j) + j^2 = 1 + 2j - 1 = 2j Next, expand j(2+j): j(2+j) = 2j + j^2 = 2j - 1 Now, add these two results: (1+j)^2 + j(2+j) = 2j + (2j - 1) = -1 + 4j Step 2: Equate the real and imaginary parts of the equation. The original equation becomes: (a+b) + j(a-b) = -1 + 4j Equating the real parts: a+b = -1 (1) Equating the imaginary parts: a-b = 4 (2) Step 3: Solve the system of linear equations for a and b. Add equation (1) and equation (2): (a+b) + (a-b) = -1 + 4 2a = 3 a = (3)/(2) Substitute the value of a into equation (1): (3)/(2) + b = -1 b = -1 - (3)/(2) b = -(2)/(2) - (3)/(2) b = -(5)/(2) The values are a=(3)/(2), b=-(5)/(2). 3 done, 2 left today. You're making progress.

Accepted Answer · 2026-05-17T14:39:54.637Z

Here are the solutions for the remaining questions. Question 11: Given z = (a+jb)/(c+jd), where a, b, c, d are real quantities. Step 1: Express z in the form X + jY by multiplying the numerator and denominator by the conjugate of the denominator. z = (a+jb)/(c+jd) × (c-jd)/(c-jd) z = ((a+jb)(c-jd))/(c^2 - (jd)^2) z = (ac - ajd + jbc - j^2bd)/(c^2 + d^2) Since j^2 = -1: z = (ac - ajd + jbc + bd)/(c^2 + d^2) Group the real and imaginary terms: z = ((ac + bd) + j(bc - ad))/(c^2 + d^2) z = (ac + bd)/(c^2 + d^2) + j(bc - ad)/(c^2 + d^2) a) If z is real, its imaginary part must be zero. Step 2: Set the imaginary part of z to zero. (bc - ad)/(c^2 + d^2) = 0 Since c^2 + d^2 ≠ 0, the numerator must be zero: bc - ad = 0 bc = ad Step 3: Rearrange the equation to show the required ratio. Assuming b ≠ 0 and d ≠ 0, divide both sides by bd: (bc)/(bd) = (ad)/(bd) (c)/(d) = (a)/(b) Thus, if z is real, then (a)/(b) = (c)/(d). b) If z is entirely imaginary, its real part must be zero. Step 4: Set the real part of z to zero. (ac + bd)/(c^2 + d^2) = 0 Since c^2 + d^2 ≠ 0, the numerator must be zero: ac + bd = 0 ac = -bd Step 5: Rearrange the equation to show the required ratio. Assuming b ≠ 0 and c ≠ 0, divide both sides by bc: (ac)/(bc) = -(bd)/(bc) (a)/(b) = -(d)/(c) Thus, if z is entirely imaginary, then (a)/(b) = -(d)/(c). Question 12: Given (a+b) + j(a-b) = (1+j)^2 + j(2+j), obtain the values of a and b. Step 1: Simplify the right-hand side of the equation. First, expand (1+j)^2: (1+j)^2 = 1^2 + 2(1)(j) + j^2 = 1 + 2j - 1 = 2j Next, expand j(2+j): j(2+j) = 2j + j^2 = 2j - 1 Now, add these two results: (1+j)^2 + j(2+j) = 2j + (2j - 1) = -1 + 4j Step 2: Equate the real and imaginary parts of the equation. The original equation becomes: (a+b) + j(a-b) = -1 + 4j Equating the real parts: a+b = -1 (1) Equating the imaginary parts: a-b = 4 (2) Step 3: Solve the system of linear equations for a and b. Add equation (1) and equation (2): (a+b) + (a-b) = -1 + 4 2a = 3 a = (3)/(2) Substitute the value of a into equation (1): (3)/(2) + b = -1 b = -1 - (3)/(2) b = -(2)/(2) - (3)/(2) b = -(5)/(2) The values are a=(3)/(2), b=-(5)/(2). 3 done, 2 left today. You're making progress.

Given z = (a+jb)/(c+jd), where a, b, c, d are real quantities, show that if z is real, then (a)/(b) = (c)/(d), and if z is entirely imaginary, then ac = -bd.

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Given z = (a+jb)/(c+jd), where a, b, c, d are real quantities, show that if z is real, then (a)/(b) = (c)/(d), and if z is entirely imaginary, then ac = -bd.

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