Ellipses & Hyperbolas — practice questions

For the ellipse $\dfrac{x^2}{9} + \dfrac{y^2}{25} = 1$, state the centre, the lengths of both axes, the coordinates of the foci, and the eccentricity.

Write $4x^2 + y^2 + 24x - 6y + 41 = 0$ in standard form. State the centre, the axes, and the foci, and verify your answer by substituting one vertex.

A satellite orbits Earth in an ellipse with the centre of the Earth at one focus. Its lowest altitude (perigee) is 500 km above the surface and its highest (apogee) is 39,000 km above the surface. Taking Earth's radius as 6,370 km, find the equation of the orbit relative to the ellipse's centre, its eccentricity, and the satellite's distance from Earth's centre when it is at the end of the minor axis.

For the hyperbola $\dfrac{y^2}{9} - \dfrac{x^2}{16} = 1$, state the centre, the vertices, the foci, the equations of the asymptotes, and the direction in which it opens.

Write $9x^2 - 4y^2 + 36x + 8y - 4 = 0$ in standard form. State its centre, vertices, foci and asymptotes, and sketch it using the box method, explaining what the box represents.

Prove that for the hyperbola $\dfrac{x^2}{a^2} - \dfrac{y^2}{b^2} = 1$ the branches approach the lines $y = \pm\dfrac{b}{a}x$ but never meet them, and use the result to explain why the rectangular hyperbola $xy = k$ has the coordinate axes as its asymptotes.