Mathematics 10 — Exercise 1.3 · Q3 (v)

Full question

For the given z, verify the displayed identity.

Step-by-step solution

1. First write z and its conjugate. Only the imaginary sign changes.

   $$(5-2\times i,2\times i+5)$$

2. Read a, the real coordinate.

   $$a=5$$

3. Read b, the coefficient of i, including its sign.

   $$b=-2$$

4. Use |z| = √(a² + b²) and substitute both coordinates.

   $$\sqrt{5^2+{\left(-2\right)}^2}$$

5. Square the real coordinate.

   $$5^2=25$$

6. Square the imaginary coordinate; a negative number squared is positive.

   $${\left(-2\right)}^2=4$$

7. Add the two squared values.

   $$25+4=29$$

8. Take the nonnegative square root of this sum.

   $$\sqrt{29}=\sqrt{29}$$

9. Repeat the modulus calculation for the comparison number.

10. Read a, the real coordinate.

    $$a=-5$$

11. Read b, the coefficient of i, including its sign.

    $$b=-2$$

12. Use |z| = √(a² + b²) and substitute both coordinates.

    $$\sqrt{{\left(-5\right)}^2+{\left(-2\right)}^2}$$

13. Square the real coordinate.

    $${\left(-5\right)}^2=25$$

14. Square the imaginary coordinate; a negative number squared is positive.

    $${\left(-2\right)}^2=4$$

15. Add the two squared values.

    $$25+4=29$$

16. Take the nonnegative square root of this sum.

    $$\sqrt{29}=\sqrt{29}$$

17. Sign changes disappear when squared, so the distances match.

    $$\sqrt{29}=\sqrt{29}$$

Answer

The final answer is shown below.