3.1.31 Curves of the 2nd order
condition: Using the selection of a complete square, bring the equation of the second order curve \( 9 x^{2}-36 x-16 y^{2}-32 y-124=0 \quad \) to canonical form. Determine the type of curve, find its semi-axes, eccentricity, coordinates of vertices and foci, equations of directrixes and asymptotes (if any). Make a drawing.
Study of curves of the second order and their characteristics. Ellipse, hyperbola, parabola. Finding eccentricities, foci, focal radii, major and minor axes, including imaginary, canonical equations and plotting.