Write the equation of the circle with center \(\mathrm{C}\) at the origin and with radius 7 .

Find the points of intersection (if any) of the circles \(\mathrm{C}_{1}\) and \(C_{2}\) where \(C_{1}: x^{2}+y^{2}-4 x-2 y+1=0\) $$ C_{2}: x^{2}+y^{2}-6 x+4 y+4=0 $$

Euclid defined a circle as the locus of points equidistant from a given point. Apollonius, on the other hand had an alternate definition: Given two points, \(\mathrm{A}\) and \(\mathrm{B}\), and a constant \(\mathrm{k} \neq 1\), the set of all points P such that \(\mathrm{PA}=\mathrm{k} \cdot \mathrm{PB}\) is a circle. Consider points \(\mathrm{A}(0,0)\) and \(\mathrm{B}(b, 0)\), and the constant \(\mathrm{k}\). Show that the Apollonian definition does indeed lead to the equation of the circle. Also, find the coordinates of the center and the radius of the circle.

For the equation \(\mathrm{x}^{2} / 25+\mathrm{y}^{2} / 9=1\), find the y coordinates when \(\mathrm{x}=2 ;\) b) \(\mathrm{x}=3 ;\) c) \(\mathrm{x}=4 ;\) d) \(\mathrm{x}=5 ;\) e) \(\mathrm{x}=6\)

Find the equation of the ellipse which has vertices \(\mathrm{V}_{1}(-2,6)\), \(\mathrm{V}^{2}(-2,-4)\), and foci $\mathrm{F}_{1}(-2,4), \mathrm{F}_{2}(-2,-2)$. (See figure,)

Given that two circles $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{D}_{1} \mathrm{x}+\mathrm{E}_{1} \mathrm{y}+\mathrm{F}_{1}=0$ and $\mathrm{x}^{2}+\mathrm{y}^{2}+\mathrm{D}_{2} \mathrm{x}+\mathrm{E}_{2} \mathrm{y}+\mathrm{F}_{2}=0$ intersect at two points, show that the equation for the line determined by the points of intersection is $\left(\mathrm{D}_{1}-\mathrm{D}_{2}\right) \mathrm{x}+\left(\mathrm{E}_{1}-\mathrm{E}_{2}\right)+\left(\mathrm{F}_{1}-\mathrm{F}_{2}\right)=0$.