Further coordinate systems

Ellipses and hyperbolas in Cartesian and parametric form, with foci, directrices and loci.

The hyperbola $H$ has equation $\dfrac{x^2}{16} - \dfrac{y^2}{9} = 1$ .

Find (a) the eccentricity of $H$ , (b) the coordinates of the foci, and (c) the equations of the directrices.

The ellipse $E$ has equation $\dfrac{x^2}{9} + \dfrac{y^2}{4} = 1$ . The point , , lies on .

M1 $\dfrac{dy}{dx} = \dfrac{dy/d\theta}{dx/d\theta} = \dfrac{2\cos\theta}{-3\sin\theta}$ (1)
M1 $y - 2\sin\theta = -\dfrac{2\cos\theta}{3\sin\theta}(x - 3\cos\theta)$ (1)
A1 Uses $\sin^2\theta + \cos^2\theta = 1$ to reach $2x\cos\theta + 3y\sin\theta = 6$ (1)
B1 $A\left(\dfrac{3}{\cos\theta}, 0\right)$ and $B\left(0, \dfrac{2}{\sin\theta}\right)$ (1)
M1 Area $= \dfrac{1}{2} \cdot \dfrac{3}{\cos\theta} \cdot \dfrac{2}{\sin\theta} = \dfrac{6}{\sin 2\theta}$ (1)
A1 Minimum when $\sin 2\theta = 1$ : minimum area $= 6$ (1)

Specification points