A line segment of length 2a has one endpoint constrained to lie on the x-axis and the other endpoint constrained to lie on the y-axis, but its endpoints are free to move along those axes. As they do so, its midpoint sweeps out a locus in the xy-plane. Obtain a rectangular coordinates equation to this locus and thereby identify this curve.
any help or ideas how to start would be appreciated. Thanx in advance.
2006-10-01 10:24:03 · 4 answers · asked by Joe 2 in Science & Mathematics ➔ Mathematics
Be sure to stop reading at the point where you want to do it yourself... because the answer is at the bottom.
This is a trig problem solvable through similar triangles and projections. First, only worry about the quadrant I solution (+x,+y). The other quadrants are just sign changes on quadrant I.
If you draw out a possible non-trivial variation of the segment - that is, one where neither endpoint is on the origin, you might notice that from the segment midpoint and its projections on the x and y axis you can form four congruent triangles. You might also notice that the x,y coordinates of the midpoint will be half of x and half of the y values of the segment endpoints.
What's left is to put one segment endpoint in terms of the other. For example: y = sqrt((2a)^2 - x^2) by the Pythagorean formula.
Next think of the midpoint projections, x' = x / 2 and y' = y / 2 and substitute them into the above.
I think you will find the resulting curve is a very familiar shape: a circle of radias a. Where x^2 + y^2 = a.
2006-10-01 11:18:06 · answer #1 · answered by Henry M 2 · 0⤊ 0⤋
The line segment makes a right triangle with the x and y axes being the sides and the hypoteneuse being the line segment.
The hypoteneuse is length 2a
Let the length of the side along the x-axis be t
Use the pythagorean theorem to find the length of the side along the y axis in terms of a and t.
The coordinate for the midpoint of the line segment will be (x/2 , y/2)
2006-10-01 10:41:38 · answer #2 · answered by Demiurge42 7 · 0⤊ 0⤋
let A be the location of one endpoint on the Oy axis
let B be the location of the other endpoint
A is of form (0, y)
B is of form (x, 0)
The parameter of the curve is the projection of segment AB on the Ox axis. For t=0 AB lies on the Oy axis.
Endpoints vary like this:
x(t)=t
y(t)=sqrt(2a-t^2)
You then calculate the rectangular coordinates eqution of the midpoint and identify the curve.
2006-10-01 10:39:30 · answer #3 · answered by Alexandra H 2 · 1⤊ 0⤋
0, 5=a a million, a million=5+bt or b=-4 0, -6=c a million, a million=-6+d or d=7 a=5, b=-4, c=-6, d=7 What you do is in basic terms plug into the equations with the numbers you recognize and discover the solutions. Soving via the substitution approach. you additionally can clean up this via determiniants
2016-12-26 06:49:56 · answer #4 · answered by ? 3 · 0⤊ 0⤋