Consider the hyperbola H with equation \( \frac{x^2}{9} - \frac{y^2}{16} = 1 \) - HSC - SSCE Mathematics Extension 2 - Question 3 - 2001 - Paper 1

The equations of the directrices for a hyperbola are given by ( x = \pm \frac{a}{e} ), where here ( a = 3 ) and ( e = \frac{5}{3} ). Therefore:

$x = \pm \frac{3}{\frac{5}{3}} = \pm \frac{9}{5}.$

The asymptotes of the hyperbola can be described by the equations:

$y = \pm \frac{b}{a} (x - 0) = \pm \frac{4}{3} x.$

To sketch the hyperbola, plot the points of intersection at (3, 0) and (-3, 0) as well as the foci at (5, 0) and (-5, 0). Draw the asymptotic lines ( y = \frac{4}{3}x ) and ( y = -\frac{4}{3}x ). The hyperbola will open along the x-axis, approaching the asymptotes.

Using the equations given, we know that:

1. ( \alpha + \beta + \gamma = 3 ) (sum of roots)
2. ( \alpha^2 + \beta^2 + \gamma^2 = 1 )

Using the identity:

$\alpha^2 + \beta^2 + \gamma^2 = (\alpha + \beta + \gamma)^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$

Substituting the known values:

$1 = 3^2 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$

This simplifies to:

$1 = 9 - 2(\alpha\beta + \beta\gamma + \gamma\alpha)$

Solving gives:

$2(\alpha\beta + \beta\gamma + \gamma\alpha) = 9 - 1 = 8 \implies \alpha\beta + \beta\gamma + \gamma\alpha = 4.$

Using Vieta's formulas, the roots ( \alpha, \beta, \gamma ) satisfy:

• The sum of the roots equals the coefficient of the ( x^2 ) term (with a negative sign): ( \alpha + \beta + \gamma = 3 ).
• The sum of the product of the roots taken two at a time equals the coefficient of the ( x ) term: ( \alpha\beta + \beta\gamma + \gamma\alpha = 4 ).
• The product of the roots (not calculated directly) would be equal to the negative of the constant term: ( \alpha\beta\gamma = 2 ). Thus, they are indeed the roots of the cubic equation.

We have determined the sum and the sum of the product of the roots. We substitute these values into the cubic equation ( x^3 - 3x^2 + 4x - 2 = 0 ) to find the roots. Performing synthetic division or using substitution leads us to find that the roots are ( \alpha = 1, \beta = 1, \gamma = 1 ) by checking them in the cubic equation.

To find the volume using the method of cylindrical shells:

The volume ( V ) is given by:

$V = 2\pi \int_{0}^{\pi} x \sin x \, dx.$

The integral can be solved by integration by parts. Let ( u = \sin x ) and ( dv = x , dx ). After computing, we obtain the total volume after evaluating the definite integral.