Question

question 2
(a) express the complex number $z = 3|45^{circ}$ in exponential form.
(2)
(b) if $z_1 = 5(\cos\frac{\pi}{2}+j\sin\frac{\pi}{2})$ and $z_2 = 4(\cos\frac{\pi}{3}+j\sin\frac{\pi}{3})$. find $\frac{z_1}{z_2}$
(3)
leave the answer in polar form.
(c) find the square roots of the complex number $x$ if $x=-3 - 4j$.
(8)
(d) write down the radius and the co - ordinates of the center of the circle
$(x - 4)^2+(y + 3)^2 = 16$
(3)
(e) write down the equation of the ellipse with center $(0;0)$, one focal point at $(0;4)$ and one vertex at $(0;5)$.
(4)
20

Explanation:

Step1: Recall exponential form of complex number

For a complex number $z = r(\cos\theta + j\sin\theta)$, its exponential form is $z = re^{j\theta}$. Given $z = 3\angle45^{\circ}=3(\cos45^{\circ}+j\sin45^{\circ})$, and $\theta = 45^{\circ}=\frac{\pi}{4}$ radians, so $z = 3e^{j\frac{\pi}{4}}$.

Step2: Divide complex numbers in polar form

If $Z_1 = r_1(\cos\theta_1 + j\sin\theta_1)$ and $Z_2 = r_2(\cos\theta_2 + j\sin\theta_2)$, then $\frac{Z_1}{Z_2}=\frac{r_1}{r_2}[\cos(\theta_1 - \theta_2)+j\sin(\theta_1 - \theta_2)]$. Here $r_1 = 5$, $\theta_1=\frac{\pi}{2}$, $r_2 = 4$, $\theta_2=\frac{\pi}{3}$. So $\frac{Z_1}{Z_2}=\frac{5}{4}[\cos(\frac{\pi}{2}-\frac{\pi}{3})+j\sin(\frac{\pi}{2}-\frac{\pi}{3})]=\frac{5}{4}(\cos\frac{\pi}{6}+j\sin\frac{\pi}{6})$.

Step3: Find square - roots of a complex number

Let $x=-3 - 4j=r(\cos\theta + j\sin\theta)$. First, find $r=\sqrt{(-3)^2+( - 4)^2}=5$, $\cos\theta=-\frac{3}{5}$, $\sin\theta=-\frac{4}{5}$, so $\theta=\pi+\arctan(\frac{4}{3})$. The square - roots of $x$ are given by $\sqrt{r}(\cos\frac{\theta + 2k\pi}{2}+j\sin\frac{\theta + 2k\pi}{2})$, $k = 0,1$.
For $k = 0$:

$$
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$$

Using the half - angle formula $\sin\frac{\alpha}{2}=\sqrt{\frac{1 - \cos\alpha}{2}}$ and $\cos\frac{\alpha}{2}=\sqrt{\frac{1+\cos\alpha}{2}}$, if $\alpha=\arctan(\frac{4}{3})$, then $\cos\alpha=\frac{3}{5}$, $\sin\alpha=\frac{4}{5}$.

$$
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$$

The square - roots are $\pm(1 - 2j)$.

Step4: Identify circle properties

The standard form of a circle is $(x - a)^2+(y - b)^2=r^2$, where $(a,b)$ is the center and $r$ is the radius. For the circle $(x - 4)^2+(y + 3)^2=16$, the center is $(4,-3)$ and the radius $r = 4$.

Step5: Find ellipse equation

For an ellipse with center $(0,0)$, $c$ is the distance from the center to a focus and $a$ is the distance from the center to a vertex. Given $c = 4$ and $a = 5$. Using the relationship $c^2=a^2 - b^2$, we can find $b^2=a^2 - c^2=25 - 16 = 9$. The equation of the ellipse is $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$.

Answer:

(a) $z = 3e^{j\frac{\pi}{4}}$
(b) $\frac{Z_1}{Z_2}=\frac{5}{4}(\cos\frac{\pi}{6}+j\sin\frac{\pi}{6})$
(c) $\pm(1 - 2j)$
(d) Radius: $4$, Center: $(4,-3)$
(e) $\frac{x^{2}}{9}+\frac{y^{2}}{25}=1$