Find the value a and b such that the cone \(ax^2 + by^2 + z^2 = 0\ \text{and the ellips}\ 2x^2 + 4y^2 + y^2 = 42\) intersect and are perpendicular to each other at the point (1, 3, 2)

Find the equation of the graph for each conic in standard form. Identify the conic, the center, the co-vertex, the focus (foci), major axis, minor axis, \(a^{2}, b^{2},\ and\ c^{2}.\) For hyperbola, find the asymptotes \(9x^{2}\ -\ 4y^{2}\ +\ 54x\ +\ 32y\ +\ 119 = 0\)

Begin with a circular piece of paper with a4-in. radius as shows in (a). cut out a sector with an arc length of x.Join the two edges of the remaining portion to form a cone with radius r and height h, as shown in (b).


a) Explain why the circumference of the bas eofthe cone is \(\displaystyle{8}{\left(\pi\right)}-{x}\).
b) Express the radius r as a function of x.
c) Express the height h as a function of x.
d) Express the volume V of the cone as afunction of x.

For Exercise,

a. Identify the equation as representing a circle, an ellipse, a hyperbola, or a parabola.

b. Graph the curve. c. Identify key features of the graph. That is. If the equation represents a circle, identify the center and radius.

If the equation represents an ellipse, identify the center, vertices, endpoints of the minor axis, foci, and eccentricity.

If the equation represents a hyperbola, identify the center, vertices, foci, equations of the asymptotes, and eccentricity.

If the equation represents a parabola, identify the vertex, focus, endpoints of the latus rectum, equation of the directrix, and equation of the axis of symmetry. \(x^2\ +\ y^2\ −\ 4x\ −\ 6y\ +\ 1 = 0\)