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

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(13200) (Show Source): You can put this solution on YOUR website!

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

The equation of the line in slope-intercept form is .

(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.

or

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).

RELATED QUESTIONS
Hi Please can somebody help with this problem i) The variables x and y are given (answered by Alan3354)

A)draw the graph of the line that passes through the point (0,-9) and has a gradient of 4 (answered by lwsshak3)

Consider the curve given by the parametric equations: x = 3t^2 - 6t y = t^4 - 2t^3 -... (answered by robertb)

A particle moves in a straight line so that its distance, s m, from a fixed point A on... (answered by Fombitz)

A particle moves in straight line so that t seconds its distance from the starting point... (answered by Alan3354)

Show by elimination that x = t^2 − 1/t^2+1 and y = 2t/ t^2+1 almost represent the unit (answered by ikleyn)

HiPlease can somebody help with this problem i) The variables x and y are given in terms... (answered by Fombitz)

Write a logarithmic equation with base 5 that is equivalent to the equation f (t) = 2t. (answered by Theo,jrfrunner)

how do you write a equation for the line that is perpendicular to the line.. for ex...y=... (answered by josgarithmetic)