91Ó°ÊÓ

Find the angle \(\theta\) (in radians and degrees) between the lines. $$\begin{aligned} &3 x-5 y=3\\\ &3 x+5 y=12 \end{aligned}$$

Find the standard form of the equation of the parabola with the given characteristics. Vertex: \((1,2) ;\) directrix: \(y=-1\)

Use the results of Exercises \(49-52\) to find a set of parametric equations to represent the graph of the line or conic. Line: passes through \((3,2)\) and \((-6,3)\)

Find the distance between the point and the line. Point \((3,2)\) Line y=2 x-1

Road Design Roads are often designed with parabolic surfaces to allow rain to drain off. A particular road that is 32 feet wide is 0.4 foot higher in the center than it is on the sides (see figure). (figure cannot copy) A. Find an equation of the parabola with its vertex at the origin that models the road surface. B. How far from the center of the road is the road surface 0.1 foot lower than in the middle?

Find the distance between the point and the line. Point \((1,-3)\) Line \(y=2 x-5\)

Conjecture Consider the ellipse \(\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1, \quad a+b=20\) (a) The area of the ellipse is given by \(A=\pi a b .\) Write the area of the ellipse as a function of \(a .\) (b) Find the equation of an ellipse with an area of 264 square centimeters. (c) Complete the table using your equation from part (a). Then make a conjecture about the shape of the ellipse with maximum area. $$\begin{array}{|l|l|l|l|l|l|l|} \hline a & 8 & 9 & 10 & 11 & 12 & 13 \\ \hline A & & & & & & \\ \hline \end{array}$$ (d) Use a graphing utility to graph the area function and use the graph to support your conjecture in part (c).

A circle and a parabola can have \(0,1,2,3,\) or 4 points of intersection. Sketch the circle \(x^{2}+y^{2}=4 .\) Discuss how this circle could intersect a parabola with an equation of the form \(y=x^{2}+C .\) Then find the values of \(C\) for each of the five cases described below. Use a graphing utility to verify your results. (a) No points of intersection (b) One point of intersection (c) Two points of intersection (d) Three points of intersection (e) Four points of intersection

The points represent the vertices of a triangle. (a) Draw triangle \(A B C\) in the coordinate plane, (b) find the altitude from vertex \(B\) of the triangle to side \(A C,\) and \((\mathrm{c})\) find the area of the triangle. $$A(-3,0), B(0,-2), C(2,3)$$

The points represent the vertices of a triangle. (a) Draw triangle \(A B C\) in the coordinate plane, (b) find the altitude from vertex \(B\) of the triangle to side \(A C,\) and \((\mathrm{c})\) find the area of the triangle. $$A(-2,0), B(0,-3), C(5,1)$$