Problem 1

In general a circle in is a curve described by an equation of the form , where , and are constants satisfying

(a). Find the Equation of the Circle, that Passes through , and

The circle has the equation .

(b). In part (a), Three Specific Points Yields a Circle. in General, what is the Minimum Number of Points Required to Determine the Equation of a Circle Uniquely? (You May Assume that the Points Are as "nice" as You Wish - no Duplicate Points, no Three Points on a Line, etc.)

Three points are required because 3 equations are required to uniquely solve a linear system involving 3 unknown variables.

(c). Circle is an Example of a Conic Section. a Conic Section in is a Curve Described by an Equation of the Form , where Are Constants and Nonzero. in General what is the Minimum Number of Points Required to Determine the Equation of a Conic Section Uniquely

Because is nonzero we can re-arrange the equation as which gives us an equation with only 5 unknown variables and thus we only need 5 points to determine a unique equation for a conic section.

Problem 2

In geometry a cubic graph is described by an equation of the form where are constants with .

(a). The Cubic Graphs that Pass through the Points and Constitute a Family of Graphs. Describe All Possible Values of and in that Family of Graphs

The family is given by any equation where and where and are any real numbers and .

(b). Do Any Graphs in the Family of Graphs in part (a) Pass through the Origin? either Demonstrate that the Answer is Yes by Finding such a Graph, or Demonstrate that the Answer is no by Proving no such Graph Exists

Yes there are multiple graphs that do one is .

Problem 3

Consider the curves and . Find all real values of for which the curves intersect at:

Since we know that or must always evaluate to 0 for this equation to be true, thus we have 3 solutions to this equation .

This gives us the equation of a circle with radius and infinitely many solutions

This gives us a system with infinitely many solutions.

This gives us a system with one solution at and

(a). Exactly on point

The curves intersect at one point for .

(b). Infinitely Many Points

The curves intersect at infinitely many points for .

(c). No Points

The curves don't intersect for any value of