SSCE HSC Mathematics Extension 2 Geometry proofs using vectors: (a)
The ellipse \( \frac{x^2}{a^

Question

(a)
The ellipse $ \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 $ has foci $ S(ae, 0) $ and $ S'(-ae, 0) $ where $ e $ is the eccentricity, with corresponding directrices $ x = \frac{a}{e} $ and $ x = -\frac{a}{e} $. The point $ P(x_0, y_0) $ lies on the ellipse. The points where the horizontal line through $ P $ meets the directrices are $ M $ and $ M' $, as shown in the diagram.

(i) Show that the equation of the normal to the ellipse at the point $ P $ is 
$$ y - y_0 = \frac{a^2}{b^2 x_0} (x - x_0) $$.

(ii) The normal at $ P $ meets the x-axis at $ N $. Show that $ N $ has coordinates $ (e^2 x_0, 0) $.

(iii) Using the focus-directrix definition of an ellipse, or otherwise, show that $ \frac{PS}{PS'} = \frac{NS}{NS'} $.

(iv) Let $ \alpha = \angle LSP'N $ and $ \beta = \angle LNP'S $. By applying the sine rule to $ \triangle LSP'N $ and to $ \triangle NPS $, show that $ \alpha = \beta $.

(b)
A light string is attached to the vertex of a smooth vertical cone. A particle $ P $ of mass $ m $ is attached to the string as shown in the diagram. The particle remains in contact with the cone and rotates with constant angular velocity $ \omega $ on a circle of radius $ r $. The string and the surface of the cone make an angle $ \alpha $ with the vertical, as shown.

(i) Resolve the forces on $ P $ in the horizontal and vertical directions.

(ii) Show that $ T = m( g \cos\alpha + r\omega^2 \sin\alpha ) $ and find a similar expression for $ T $.

(a) The ellipse \( \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1 \) has foci \( S(ae, 0) \) and \( S'(-ae, 0) \) where \( e \) is the eccentricity, with corresponding directrices \( x = \frac{a}{e} \) and \( x = -\frac{a}{e} \) - HSC - SSCE Mathematics Extension 2 - Question 4 - 2009 - Paper 1

Show that the equation of the normal to the ellipse at the point P is \( y - y_0 = \frac{a^2}{b^2 x_0} (x - x_0) \)

Show that N has coordinates \( (e^2 x_0, 0) \)

Show that \( \frac{PS}{PS'} = \frac{NS}{NS'} \)

By applying the sine rule to \( \triangle LSP'N \) and to \( \triangle NPS \), show that \( \alpha = \beta \)

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