The complex number $z = 1 - 4i$, where $i^2 = -1$. (a) Plot $z$ and $-2z$ on the Argand diagram. (b) Show that $2 |z| = |z| = |-2z|$. (c) What does part (b) tell you about the points you plotted in part (a)? (d) Let $k$ be a real number such that $|z + k| = 5$. Find the two possible values of $k$.

Leaving Cert Mathematics Complex Numbers: The complex number $z = 1

The complex number $z = 1 - 4i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 3 - 2012

Question 3

The complex number $z = 1 - 4i$, where $i^2 = -1$. (a) Plot $z$ and $-2z$ on the Argand diagram. (b) Show that $2 |z| = |z| = |-2z|$. (c) What does part (b) tell ... show full transcript

Worked Solution & Example Answer:The complex number $z = 1 - 4i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 3 - 2012

Step 1

Plot $z$ and $-2z$ on the Argand diagram.

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Answer

To plot the complex number $z = 1 - 4i$ , locate the point at $(1, -4)$ on the Argand diagram. For $-2z$ , calculate:

$-2z = -2(1 - 4i) = -2 + 8i$

This corresponds to the point $(-2, 8)$ . Thus, plot the points at $(1, -4)$ and $(-2, 8)$ on the diagram.

Step 2

Show that $2 |z| = |z| = |-2z|$.

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Answer

First, calculate the modulus of $z$ :

$|z| = |1 - 4i| = \\sqrt{1^2 + (-4)^2} = \\sqrt{1 + 16} = \\sqrt{17}$

Now calculate $|-2z|$ :

$|-2z| = |-2(1 - 4i)| = |2 - 8i| = \\sqrt{(-2)^2 + (-8)^2} = \\sqrt{4 + 64} = \\sqrt{68} = 2\sqrt{17}$

Therefore, we have:

$2|z| = 2 \\cdot \\sqrt{17} = |z| + |(-2z)|$

Step 3

What does part (b) tell you about the points you plotted in part (a)?

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Answer

-2z is twice as far from the origin as $z$ is. This indicates that $-2z$ is located further away from the origin in the Argand diagram compared to $z$ .

Step 4

Let $k$ be a real number such that $|z + k| = 5$. Find the two possible values of $k$.

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Answer

Start with:

$|z + k| = |1 - 4i + k| = 5$

This can be rearranged to:

$|k + 1 - 4i| = 5$

Calculate the modulus:

$\sqrt{(k + 1)^2 + (-4)^2} = 5$

Squaring both sides gives:

$(k + 1)^2 + 16 = 25$

This simplifies to:

$(k + 1)^2 = 9$

Taking square roots on both sides:

$k + 1 = 3 \\text{ or } k + 1 = -3$

Thus, we find:

$k = 2 \\text{ or } k = -4$

The complex number $z = 1 - 4i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 3 - 2012

Worked Solution & Example Answer:The complex number $z = 1 - 4i$, where $i^2 = -1$ - Leaving Cert Mathematics - Question 3 - 2012

Plot $z$ and $-2z$ on the Argand diagram.

Show that $2 |z| = |z| = |-2z|$.

What does part (b) tell you about the points you plotted in part (a)?

Let $k$ be a real number such that $|z + k| = 5$. Find the two possible values of $k$.

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