The base of a solid is the region enclosed by the parabola $x = 4 - y^2$ and the $y$-axis - HSC - SSCE Mathematics Extension 2 - Question 6 - 2009 - Paper 1

To show that $\frac{1}{\alpha}$ is a zero, we substitute $\frac{1}{\alpha}$ into $P(x)$. Since $\alpha$ is a root, we have:

$P(\alpha) = 0 \Rightarrow (\alpha)^3 + q(\alpha^2) + q(\alpha) + 1 = 0.$

Multiplying each term by $\frac{1}{\alpha^3}$ results in:

$1 + \frac{q}{\alpha} + \frac{q}{\alpha^2} + \frac{1}{\alpha^3} = 0,$

which rearranges to show that $P(\frac{1}{\alpha}) = 0$. Thus, $\frac{1}{\alpha}$ is indeed a root.

Given that $\alpha$ is a root of $P(x)$ and is not real, we utilize the properties of complex roots in polynomials. If $\alpha$ is not real, its conjugate $\bar{\alpha}$ must also be a root. By Vieta's formulas, the product of the roots will equal $(-1)$, thus:

$\alpha \cdot \bar{\alpha} = |\alpha|^2 = 1 \Rightarrow |\alpha| = 1.$

Using Vieta's formulas again, for the polynomial $P(x)$ where the sum of the roots equates to $-q$ and knowing one of the roots is $\alpha$ and the other two can be expressed in terms of $\bar{\alpha}$, we can set:

$\alpha + \bar{\alpha} + other ext{{ root}} = -q.$

With the modulus equal to 1, we conclude:

$Re(\alpha) + Re(\bar{\alpha}) = -q \Rightarrow 2 Re(\alpha) = -q \Rightarrow Re(\alpha) = \frac{1 - q}{2}.$

To find the length of $PQ$, we note that $PQ$ is tangent to the circle at point $Q$. The distance from point $O$ to point $Q$ is given as the radius $r$. Using the coordinates of point $P(x, y)$, the distance formula yields:

$PQ = \sqrt{(x - c)^2 + (y - \sqrt{r^2 - c^2})^2}.$

From the condition of equality $PQ = PR$, we can express both distances in terms of coordinates leading us to manipulate the resulting equations. The equation of the circle is represented as $x^2 + y^2 = r^2$. The tangential distance gives rise to a quadratic expression which simplifies to:

$y^2 = r^2 + c^2 - 2cK,$

providing us the locus of point $P$.

To find the focus of the parabola defined by the equation derived in part (ii), we utilize the standard form for a parabola $y^2 = 4p(x - h)$ where $(h, k)$ is the vertex. The distance $p$ is related to the coefficients defining the parabola; accordingly, an analysis of congruence with the derived formula would yield coordinates for focus $S$. Specifically, if the vertex lies on $c$, the focus lies $p$ units from this vertex along the axis of symmetry.

To analyze the difference in lengths, we express these in terms of their geometric representations. By contrasting distances $PS$ and $PQ$, we can ascertain the relationship remains constant under transformations with respect to $x$. A simple application of derivative forms or geometric mean arguments will illustrate the independence of $x$, confirming the lengths differ consistently across various positions of $P$.