
A complex number is given such that $x - iy = {( - 7 - 24i)^{\dfrac{1}{2}}}$ . What is the value of ${x^2} + {y^2}$ ?
A. 15
B. 25
C. -25
D. None of these
Answer
164.4k+ views
Hint: Iota, $i$ is a complex number such that $\sqrt i = - 1$ which gives us ${i^2} = - 1$ . Now, to solve the above question, use a relation which connects both ${({x^2} + {y^2})^2}$ and ${({x^2} - {y^2})^2}$ tomake the calculations further easier in evaluating the value of $({x^2} + {y^2})$ .
Complete step by step solution:
The given number is: $x - iy = \sqrt {( - 7 - 24i)} $
On squaring both the sides, we get
${(x - iy)^2} = - 7 - 24i$
Opening the square,
${x^2} + {(iy)^2} - 2ixy = - 7 - 24i$
Now, the property of the complex number iota, $i$ is that ${i^2} = - 1$ .
Thus, proceeding further
${x^2} - {y^2} - 2ixy = - 7 - 24i$
On comparing both, the real part, and the imaginary part on both the sides we get:
${x^2} - {y^2} = - 7$ … (1)
And
$2xy = 24$ … (2)
Now, we will try to write $({x^2} + {y^2})$ in terms of $({x^2} - {y^2})$ .
We know that
${({x^2} + {y^2})^2} = {x^4} + {y^4} + 2{x^2}{y^2}$
Adding and subtracting $4{x^2}{y^2}$ on the right side of the above equation,
${({x^2} + {y^2})^2} = {x^4} + {y^4} + 2{x^2}{y^2} + 4{x^2}{y^2} - 4{x^2}{y^2}$
Simplifying further,
${({x^2} + {y^2})^2} = {x^4} + {y^4} - 2{x^2}{y^2} + 4{x^2}{y^2}$
As we know that $({x^2} - {y^2}) = {x^4} + {y^4} - 2{x^2}{y^2}$ . Hence, on substituting this we get:
${({x^2} + {y^2})^2} = {({x^2} - {y^2})^2} + 4{x^2}{y^2}$
Simplifying further,
${({x^2} + {y^2})^2} = {({x^2} - {y^2})^2} + {(2xy)^2}$
Substituting the values of $({x^2} - {y^2})$ from equation (1) and $(2xy)$ from equation (2) in the above equation,
${({x^2} + {y^2})^2} = {( - 7)^2} + {(24)^2} = 49 + 576$
Therefore, ${({x^2} + {y^2})^2} = 625$ .
Taking the square root, we get:
${x^2} + {y^2} = \pm 25$
As the sum of the squares of two numbers is always positive, therefore, ${x^2} + {y^2} = 25$ .
Thus, the correct option is B.
Note: On evaluating the square root of a variable, its value can be both positive and negative. For example, let’s say that ${x^2} = {a^2}$ . Then, on taking the square root one should always remember that $x = \pm a$ both. On analyzing the properties of $x$ , one can also further evaluate whether its value is positive and negative, just like we did in the question above.
Complete step by step solution:
The given number is: $x - iy = \sqrt {( - 7 - 24i)} $
On squaring both the sides, we get
${(x - iy)^2} = - 7 - 24i$
Opening the square,
${x^2} + {(iy)^2} - 2ixy = - 7 - 24i$
Now, the property of the complex number iota, $i$ is that ${i^2} = - 1$ .
Thus, proceeding further
${x^2} - {y^2} - 2ixy = - 7 - 24i$
On comparing both, the real part, and the imaginary part on both the sides we get:
${x^2} - {y^2} = - 7$ … (1)
And
$2xy = 24$ … (2)
Now, we will try to write $({x^2} + {y^2})$ in terms of $({x^2} - {y^2})$ .
We know that
${({x^2} + {y^2})^2} = {x^4} + {y^4} + 2{x^2}{y^2}$
Adding and subtracting $4{x^2}{y^2}$ on the right side of the above equation,
${({x^2} + {y^2})^2} = {x^4} + {y^4} + 2{x^2}{y^2} + 4{x^2}{y^2} - 4{x^2}{y^2}$
Simplifying further,
${({x^2} + {y^2})^2} = {x^4} + {y^4} - 2{x^2}{y^2} + 4{x^2}{y^2}$
As we know that $({x^2} - {y^2}) = {x^4} + {y^4} - 2{x^2}{y^2}$ . Hence, on substituting this we get:
${({x^2} + {y^2})^2} = {({x^2} - {y^2})^2} + 4{x^2}{y^2}$
Simplifying further,
${({x^2} + {y^2})^2} = {({x^2} - {y^2})^2} + {(2xy)^2}$
Substituting the values of $({x^2} - {y^2})$ from equation (1) and $(2xy)$ from equation (2) in the above equation,
${({x^2} + {y^2})^2} = {( - 7)^2} + {(24)^2} = 49 + 576$
Therefore, ${({x^2} + {y^2})^2} = 625$ .
Taking the square root, we get:
${x^2} + {y^2} = \pm 25$
As the sum of the squares of two numbers is always positive, therefore, ${x^2} + {y^2} = 25$ .
Thus, the correct option is B.
Note: On evaluating the square root of a variable, its value can be both positive and negative. For example, let’s say that ${x^2} = {a^2}$ . Then, on taking the square root one should always remember that $x = \pm a$ both. On analyzing the properties of $x$ , one can also further evaluate whether its value is positive and negative, just like we did in the question above.
Recently Updated Pages
Environmental Chemistry Chapter for JEE Main Chemistry

Geometry of Complex Numbers – Topics, Reception, Audience and Related Readings

JEE Main 2021 July 25 Shift 1 Question Paper with Answer Key

JEE Main 2021 July 22 Shift 2 Question Paper with Answer Key

JEE Atomic Structure and Chemical Bonding important Concepts and Tips

JEE Amino Acids and Peptides Important Concepts and Tips for Exam Preparation

Trending doubts
JEE Main 2025 Session 2: Application Form (Out), Exam Dates (Released), Eligibility, & More

Atomic Structure - Electrons, Protons, Neutrons and Atomic Models

Displacement-Time Graph and Velocity-Time Graph for JEE

JEE Main 2025: Derivation of Equation of Trajectory in Physics

Learn About Angle Of Deviation In Prism: JEE Main Physics 2025

Electric Field Due to Uniformly Charged Ring for JEE Main 2025 - Formula and Derivation

Other Pages
JEE Advanced Marks vs Ranks 2025: Understanding Category-wise Qualifying Marks and Previous Year Cut-offs

JEE Advanced Weightage 2025 Chapter-Wise for Physics, Maths and Chemistry

NCERT Solutions for Class 11 Maths Chapter 4 Complex Numbers and Quadratic Equations

Degree of Dissociation and Its Formula With Solved Example for JEE

Instantaneous Velocity - Formula based Examples for JEE

JEE Main 2025: Conversion of Galvanometer Into Ammeter And Voltmeter in Physics
