SOLUTION: Graph the line that is described parametrically by (x,y) = (2t, 5-t), then: (a) Confirm that the point corresponding to t=0 is exactly 5 units from (3,9); (b) Write a formula i

Algebra ->  Equations -> SOLUTION: Graph the line that is described parametrically by (x,y) = (2t, 5-t), then: (a) Confirm that the point corresponding to t=0 is exactly 5 units from (3,9); (b) Write a formula i      Log On


   

Question 1196068: Graph the line that is described parametrically by (x,y) = (2t, 5-t), then:
(a) Confirm that the point corresponding to t=0 is exactly 5 units from (3,9);
(b) Write a formula in terms of t for the distance from (3,9) to (2t, 5-t)
(c) Find one other point on the line that is 5 units from (3,9)
(d) Find the point on the line that minimizes the distance to (3,9)
Answer by greenestamps(13198) About Me  (Show Source):
You can put this solution on YOUR website!


Given x=2t, find y in terms of x:

y=5-t=5-%281%2F2%29%282t%29=5-%281%2F2%29x

The equation of the line in slope-intercept form is y=%28-1%2F2%29x%2B5.

graph%28400%2C200%2C-2%2C12%2C-1%2C6%2C%28-1%2F2%29x%2B5%29

(a) The point corresponding to t=0 is (0,5). From that point to (3,9) the difference in x is 3 and the difference in y is 4, so the distance to (3,9) is 5 (by the 3-4-5 Pythagorean Triple).

(b) To avoid the confusion of the square root, I will find the formula for the square of the distance from (3,9) to a point on the line.

d%5E2=%283-2t%29%5E2%2B%289-%285-t%29%29%5E2=4t%5E2-12t%2B9%2Bt%5E2%2B8t%2B16=5t%5E2-4t%2B25

(c) Find where the square of the distance is 5^2=25:

5t%5E2-4t%2B25=25
5t%5E2-4t=0
t%285t-4%29=0

t=0 or t=4%2F5

We already knew t=0 was one solution; t=4/5 is the other. The point on the line corresponding to t=4/5 is (2t,5-t) = (8/5,21/5).

CHECK: (3-8/5)^2+(9-21/5)^2 = (7/5)^2+(24/5)^2 = 49/25+576/25 = 625/25 = 25

(d) By symmetry, the point on the line that minimizes the distance from (3,9) is halfway between the two points that are the same distance 5 from the line. That corresponds to t halfway between 0 and 4/5, or t = 2/5. That point on the line is (2t,5-t) = (4/5,23/5).