M203 20251213 Conics_Maxima, Minima of Quadratics_Circles_Ellipses_Hyperbolas
2 . Maxima and Minima of Quadratics
2 ) Let $f(x)=ax^2+bx+c$, where $a,b,c$ are real numbers and $a>0$. In terms of $a,b$ and $c$, find the real value of $x$ that makes $f(x)$ as small as possible. What happens if $a$ is negative?
3 ) Let $k$ be a real number. Find the minimum possible distance between $k(1+3i)$ and $2i$ in the complex plane.
7 ) A) Find the minimum value of $p(y)=2y^2-4y+19$
B) Find the maximum value of $37-16r-r^2$
6 ) Suppose the temperature of each point $(x,y)$ in the plane is given by the function $x^2+y^2-4x+6y$. Find the coldest point in the plane and determine its temperature.
3 . Circles
1 . Find the distance from $(5,7)$ to the nearest point on the graph of $2x^2+4x +2y^2 – 8y = 6$
4 . Find an equation whose graph is a circle that has a diameter with endpoints $(3,2)$ and $(7,-4)$.
(5,-1) $r= \dfrac{\sqrt{52}}{2}$ $r^2 = \dfrac{52}{2} = 13$
9 . The circumcircle of a triangle is the circle that passes through all three vertices of the triangle. Find an equation whose graph is the circumcircle of a triangle with vertices $(-2,5)(-4,-3)(0,-3)$
4 . Ellipse
2 ) Graph each of the following ellipses below, and find an equation that produces the given graph. Find the center, foci and the lengths of
both axes of each ellipse.
a. The center is $(4, -2)$, the major axis is horizontal with length $6$, and $(4, -3)$ is one endpoint of the minor axis.
3 ） Find the center, foci, and the lengths of both axes of each graph.
a. $\dfrac{(x-1)^2}{16} + \dfrac{(y+7)^2}{25} =1$
(1,-7), foci major axis 10 minor axis 8 , foci (1,-4) (1,-10)
7 ) An ellipse has foci (0, 0) and (14, 0) and passes through the vertex of the parabola with equation $y = x^2 – 10x + 37$. Find the length of the major and minor axis of this ellipse and the distance of foci.