Let $ heta$ be the complex number satisfying $ar{ heta}^3 = 1$ and $ ext{Im}( heta) > 0$ - HSC - SSCE Mathematics Extension 2 - Question 6 - 2008 - Paper 1

To find the equation of the tangent line at point $P$:

1. Differentiate the hyperbola equation to find the slope at $P$.
2. Use the point-slope form of the line equation.
3. By substituting the coordinates of point $P$, we derive the equation to be:

$bx ext{ sec } heta - ay an heta - ab = 0$.

To show the length of segment $SR$:

1. Apply the Pythagorean theorem.
2. Use trigonometric identities to express the distances in terms of $a$, $b$, and $heta$.
3. Simplify to arrive at:

SR = rac{ab( ext{sec } heta - 1)}{rac{1}{ an^2 heta} + b^2 ext{sec}^2 heta}.

To demonstrate this relation, we apply the properties of the hyperbola and the lengths calculated before:

1. Express both $SR$ and $S'R'$ in terms of $b$ and simplify.
2. Verify the product results in $b^2$.

To prove this identity:

1. Start with the left-hand side: rac{1}{r} - rac{r-1}{r} = rac{-1}{r}
2. Rearranging gives rac{1}{n}. Thus proving the equality.

To demonstrate this:

1. Simplify the left-hand side by adding values.
2. After rearranging, you end up with the relationship:

rac{1}{n}.

As $m$ increases:

1. The term rac{1}{inom{m}{n}} approaches $0$ when considering the growth of binomial coefficients for large $m$. Therefore, the limiting value is $0$.