Taxicab Geometry 2: Ellipses and Hyperbolas

Name: Honors Math 3 problem set Date: Investigation 9A extension (continued) Taxicab Geometry 2: ellipses and hyperbolas

Warm-up: Euclidean ellipses and hyperbolas For problems 12 and 13, use the ordinary definition of distance from Euclidean geometry. Make your graphs on graph paper. Use a ruler and a compass as needed to make your graphs reasonabl y accurate, or use focus paper instead. 12. Suppose P = (4, 2) and Q = (1, 9). Draw the ellipse that has P and Q as foci and 14 as the sum of the focal distances. That is, graph all points X having the property that PX + QX = 14. Hint: You can find some of the points by intersecting the circle where PX = 10 and the circle w here QX = 4. Repeat with other pairs of numbers adding to 14. 13. Suppose P = (4, 2) and Q = (1, 9). Draw the hyperbola that has P and Q as foci and 2 as the d ifference of the focal distances. That is, graph all points X having the property that | PX – QX | = 2.

Drawing a taxicab ellipse and a taxicab hyperbola For the remainder of this assignment, all distances are taxicab distances (as defined in the first ha ndout on Taxicab Geometry) and circle always refers to a taxicab circle. As in some of the problems from that handout, suppose there’s a city in which Paula lives at P = (4, 2) and Quincy lives at Q = (1, 9). 14. For the points P = (4, 2) and Q = (1, 9), follow these steps to draw a taxicab ellipse that has P and Q as foci and 14 as the sum of the focal distances. Do all parts of this problem on the s ame graphing grid.

a. Draw the circle of points X having dT(P, X) = 10 and draw the circle of points X having dT (Q, X) = 4. Mark the points where these circles intersect.

b. Draw the circle of points X having dT(P, X) = 8 and draw the circle of points X having dT (Q, X) = 6. Mark the points where these circles intersect.

c. Draw the circle of points X having dT(P, X) = 7 and draw the circle of points X having dT (Q, X) = 7. Mark the points where these circles intersect.

d. Draw the circle of points X having dT(P, X) = 12 and draw the circle of points X having dT (Q, X) = 2. Mark the points where these circles intersect.

e. Draw the circle of points X having dT(P, X) = 13 and draw the circle of points X having dT (Q, X) = 1. Mark the points where these circles intersect. f. Repeat the previous process for several other pairs of numbers that add to 14. Then, use a ll the intersections you have found to draw the taxicab ellipse described by this equation: dT(P, X) + dT(Q, X) = 14. g. Write a story problem about Paula and Quincy whose solution would be the taxicab ellips e drawn in part f. 15. Paula and Quincy like to hang out at a park that is two blocks further from one of their homes than it is from the other of their homes. Following a process similar to the previous problem, draw the taxicab hyperbola showing all the possible locations of the park. That is, graph the points X for which | dT(P, X) – dT(Q, X) | = 2.