Question 694263: How do you design a nonlinear system that has at least two solutions and one solution must be the ordered pair, (-2, 5)? Can you please tell me how you came up with your system and give the entire solution set for the system. Thank you in advance.
Answer by solver91311(24713)   (Show Source):
You can put this solution on YOUR website!
There are infinite solutions to this problem, however the following should suffice.
Assume you have a circle that contains the given point, (-2,5). Further assume that the center of this circle is the origin. Using the distance formula, we find the radius of this circle to be ^2\ +\ (5\ -\ 0)^2}\ =\ \sqrt{29})
Then using the pattern for the equation of a circle of radius  centered at the origin,  , we write the equation of the required circle:
Now assume a parabola with a vertex on the  -axis and that contains the point (-2,5). By symmetry, such a parabola must also pass through the point (2,5). I leave it as an exercise for the student to convince himself of the fact that this other point is, indeed, on both the parabola and the circle.
Just for the sake of simplicity of arithmetic, let's choose (0,-3) as the vertex for the parabola. In fact, any point on the -axis would do, but this particular point results in tidy arithmetic.
Now that we have three non-collinear points, we can write the equation of a parabola that contains all three points. Note that the general form of a parabola is:
Since the particular parabola for which we want an equation must contain the point (-2,5). That is to say, if  , then  . In other words,
^2\ +\ b(-2)\ +\ c\ =\ 5)
or more simply put:
Similarly, since the parabola must contain the points (2, 5) and (0, -3),
and
which is to say
Solving this system of equations to obtain the coefficients for the parabola equation is left as an exercise for the student.
Obviously, the solution set is the two points that the circle and the parabola have in common, namely (-2,5) and (2,5).
Super Double Plus Extra Credit: Find a point for the vertex of the parabola that results in 3 points in the solution set of the system. Find another point for the vertex that results in 4 points in the solution set.
John
Egw to Beta kai to Sigma
My calculator said it, I believe it, that settles it
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