16. Parametric Equations & Polar Coordinates

Determine if the equation $x^3+y^2+4x-8y+4=0$ is a circle, and if it is, find its center and radius.

Given the equation $\(\frac{x^2}{4}\)+\(\frac{y^2}{9}\)=1$, sketch a graph of the ellipse.

Given the ellipse equation $\(\frac{x^2}{16}\)+\(\frac{y^2}{4}\)=1$, determine the magnitude of the semi-major axis (a) and the semi-minor axis (b).

Determine the vertices and foci of the following ellipse: $\(\frac{x^2}{49}\)+\(\frac{y^2}{36}\)=1$.

Determine the vertices and foci of the following ellipse: $\(\frac{x^2}{9}\)+\(\frac{y^2}{16}\)=1$.

Find the standard form of the equation for an ellipse with the following conditions.

Foci = $\(\left\)(-5,0\(\right\)),\(\left\)(5,0\(\right\))$

Vertices = $\(\left\)(-8,0\(\right\)),\(\left\)(8,0\(\right\))$

Graph the ellipse $\(\frac{\left(x-1\right)^2}{9}\)+\(\frac{\left(y+3\right)^2}{4}\)=1$.

Determine the vertices and foci of the ellipse $\(\left\)(x+1\(\right\))^2+\(\frac{\left(y-2\right)^2}{4}\)=1$.

Graph the parabola $-4\(\left\)(y+1\(\right\))=\(\left\)(x+1\(\right\))^2$, and find the focus point and directrix line.

If a parabola has the focus at $\(\left\)(0,-1\(\right\))$ and a directrix line $y=1$, find the standard equation for the parabola.

Graph the parabola $8\(\left\)(x+1\(\right\))=\(\left\)(y-2\(\right\))^2$ , and find the focus point and directrix line.

If a parabola has the focus at $\(\left\)(2,4\(\right\))$ and a directrix line $x=-4$ , find the standard equation for the parabola.

Given the hyperbola $\(\frac{x^2}{25}\)-\(\frac{y^2}{9}\)=1$, find the length of the $a$-axis and the $b$-axis.

Given the hyperbola $x^2-\(\frac{y^2}{4}\)=1$, find the length of the $a$-axis and $b$-axis.

Given the hyperbola $\(\frac{y^2}{100}\)-\(\frac{x^2}{139}\)=1$, find the length of the $a$-axis and the $b$-axis.