91Ó°ÊÓ

The graph of the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with \(a>0, b>0\) is a hyperbola with _______________ (horizontal/vertical) transverse axis, vertices (___, ___) and (___, ___) and foci \((\pm c, 0),\) where \(c=\) _______________ . So the graph of \(\frac{x^{2}}{4^{2}}-\frac{y^{2}}{3^{2}}=1\) is a hyperbola with vertices (___, ___) and (___, ___) and foci (___, ___) and (___, ___).

An equation of a parabola is given. (a) Find the focus, directrix, and focal diameter of the parabola. (b) Sketch a graph of the parabola and its directrix. $$y=-\frac{1}{8} x^{2}$$

Identifying a Parabola Using Rotation of Axes (a) Use rotation of axes to show that the following equation represents a parabola. $$2 \sqrt{2}(x+y)^{2}=7 x+9 y$$ (b) Find the \(X Y\) - and \(x y\) -coordinates of the vertex and focus. (c) Find the equation of the directrix in \(X Y\) - and \(x y\) -coordinates.

The polar equation of an ellipse can be expressed in terms of its eccentricity \(e\) and the length \(a\) of its major axis. (a) Show that the polar equation of an ellipse with directrix \(x=-d\) can be written in the form $$r=\frac{a\left(1-e^{2}\right)}{1-e \cos \theta}$$ [Hint: Use the relation \(a^{2}=e^{2} d^{2} /\left(1-e^{2}\right)^{2}\) given in the proof on page 825 .] (b) Find an approximate polar equation for the elliptical orbit of the earth around the sun (at one focus) given that the eccentricity is about 0.017 and the length of the major axis is about \(2.99 \times 10^{8} \mathrm{km}\).

Finding the Equation of an Ellipse Find an equation for the ellipse that satisfies the given conditions. Length of major axis: \(6,\) length of minor axis: \(4,\) foci on \(x\) -axis

The planets move around the sun in elliptical orbits with the sun at one focus. The point in the orbit at which the planet is closest to the sun is called perihelion, and the point at which it is farthest is called aphelion. These points are the vertices of the orbit. The earth's distance from the sun is \(147,000,000 \mathrm{km}\) at perihelion and \(153,000,000 \mathrm{km}\) at aphelion. Find an equation for the earth's orbit. (Place the origin at the center of the orbit with the sun on the \(x\) -axis.) (IMAGE CAN'T COPY)