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If x + i y =sqrt((a+i b)/(c+i d)) prove ...

If `x + i y =sqrt((a+i b)/(c+i d))` prove that `(x^2+y^2)^2=(a^2+b^2)/(c^2+d^2)`

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AI Generated Solution

To prove that \((x^2 + y^2)^2 = \frac{a^2 + b^2}{c^2 + d^2}\) given that \(x + iy = \sqrt{\frac{a + ib}{c + id}}\), we can follow these steps: ### Step 1: Start with the given equation We have: \[ x + iy = \sqrt{\frac{a + ib}{c + id}} \] ...
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Knowledge Check

  • If x = iy = sqrt((a + i b)/( c + id)) , "then " (x^(2) + y^(2))^(2) is equal to

    A
    ` (a^(2) + b^(2))/(c^(2) + d^(2))`
    B
    `(c^(2) + d^(2))/(a^(2) + b^(2))`
    C
    `(a^(2) - b^(2))/(c^(2) - d^(2))`
    D
    none
    • NCERT-COMPLEX NUMBERS AND QUADRATIC EQUATIONS-MISCELLANEOUS EXERCISE
    1. If x + i y =sqrt((a+i b)/(c+i d)) prove that (x^2+y^2)^2=(a^2+b^2)/(c^...

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    2. Find the number of non-zero integral solutions of the equation |1-i|^...

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    3. If (a + i b) (c + i d) (e + if) (g + i h) = A + i B, then show that (a...

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    4. Let z1=2-i ,z2=-2+i. Find Re ((z1z2)/( bar z'1))

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    5. Find the modulus and argument of the complex number (1+2i)/(1-3i).

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    6. If z1=2-i ,z2=1+i ,find |(z1+z2+1)/(z1-z2+i)|

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    7. If a + i b =((x+i)^2)/(2x^2+1),prove that a^2+b^2=((x^2+1)^2)/((2x^2+...

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    8. If alphaand betaare different complex numbers with |beta|=1,then fin...

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    9. Find the real numbers x and y if (x-i y)(3+5i) is the conjugate of -6...

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    10. Find the modulus of (1+i)/(1-i)-(1-i)/(1+i).

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    11. If (x+i y)^3=u+i v ,then Find p if u/x+v/y=p(x^2-y^2).

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    12. Convert the following in the polar form : (i) (1+7i)/((2-i)^2) (ii) ...

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    13. Solve the equation :x^2-2x+3/2=0

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    14. Solve the equation : 3x^2-4x+(20)/3=0

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    15. Evaluate : [i^(18)+(1/i)^(25)]^3

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    16. Reduce (1/(1-4i)-2/(1+i))((3-4i)/(5+i))to the standard form.

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    17. For any two complex numbers z1and z2, prove that Re(z1z2)=Re(z1) Re(z2...

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    18. Solve the equation :21 x^2-28 x+10=0

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    19. Solve the equation :27 x^2-10 x+1=0

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    20. If ((1+i)/(1-i))^m=1,then find the least integral value of m.

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