91Ó°ÊÓ

Find a polar equation of the conic with a focus at the pole and the given eccentricity and directrix. $$e=3, r=3 \csc \theta$$

The lighting of the National Christmas Tree located on the Ellipse, a large grassy area south of the White House, marks the beginning of the holiday season in Washington, D.C. This area of the lawn is actually an ellipse with major axis of length \(1048 \mathrm{ft}\) and minor axis of length 898 ft. Assuming that a coordinate system is superimposed on the area in such a way that the center is at the origin and the major and minor axes are on the \(x\) - and \(y\) -axes of the coordinate system, respectively, find an equation of the ellipse.

In Exercises \(55-58,\) determine whether the statement is true or false. A nonlinear system of equations can have both realnumber solutions and imaginary-number solutions.

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { piecewise function } & \text { ellipse }\\\ \text { linear equation } & \text { midpoint } \\ \text { factor } & \text { distance } \\ \text { remainder } & \text { one real-number } \\ \text { solution } & \text { solution } \\ \text { zero } & \text { two different real-number } \\\ x \text { -intercept } & \text { solutions } \\ y \text { -intercept } & \text { two different imaginary- } \\ \text { parabola } & \text { number solutions } \\ \text { circle } & \end{array}$$ The ______________________ between two points \(\left(x_{1}, y_{1}\right)\) and \(\left(x_{2}, y_{2}\right)\) is given by \(\left(\frac{x_{1}+x_{2}}{2}, \frac{y_{1}+y_{2}}{2}\right)\).

Fill in the blank with the correct term. Some of the given choices will not be used. $$\begin{array}{ll}\text { piecewise function } & \text { ellipse }\\\ \text { linear equation } & \text { midpoint } \\ \text { factor } & \text { distance } \\ \text { remainder } & \text { one real-number } \\ \text { solution } & \text { solution } \\ \text { zero } & \text { two different real-number } \\\ x \text { -intercept } & \text { solutions } \\ y \text { -intercept } & \text { two different imaginary- } \\ \text { parabola } & \text { number solutions } \\ \text { circle } & \end{array}$$ For a quadratic equation \(a x^{2}+b x+c=0,\) if \(b^{2}-4 a c>0,\) the equation has ___________________ .

Graph the system of inequalities. Then find the coordinates of the points of intersection of the graphs of the related equations. $$\begin{aligned} &y \geq x^{2}\\\ &y

Numerical Relationship. The sum of two numbers is \(1,\) and their product is \(1 .\) Find the sum of their cubes. There is a method to solve this problem that is easier than solving a nonlinear system of equations. Can you discover it?

Box Dimensions. Four squares with sides 5 in. long are cut from the corners of a rectangular metal sheet that has an area of 340 in \(^{2}\). The edges are bent up to form an open box with a volume of 350 in \(^{3}\). Find the dimensions of the box.