SOLUTION: Find an equation of the set of points in a plane such that the sum of distances between each point of the set and the points (1,0) and (-1,0) is 8 units

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Question 1033012: Find an equation of the set of points in a plane such that the sum of distances between each point of the set and the points (1,0) and (-1,0) is 8 units
Found 2 solutions by josgarithmetic, ikleyn:
Answer by josgarithmetic(39617) About Me  (Show Source):
You can put this solution on YOUR website!
The description of an ellipse with foci (1,0) and (-1,0).

sqrt%28%28x-1%29%5E2%2B%28y-0%29%5E2%29%2Bsqrt%28%28x-%28-1%29%29%5E2%2B%28y-0%29%5E2%29=8, using Distance Formula. Simplify the equation. Adjust into Standard Form.

Answer by ikleyn(52786) About Me  (Show Source):
You can put this solution on YOUR website!
.
Find an equation of the set of points in a plane such that the sum of distances between each point
of the set and the points (1,0) and (-1,0) is 8 units.
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It is an ellipse, and it is its characteristic property.


Read about it in the lesson Ellipse focal property in this site.

The foci of this ellipse are the points (1,0) and (-1,0). 

The major semi-axes of the ellipse a = 8%2F2 = 4 is half of the sum of these distances. 

The minor semi-axes b = sqrt%28a%5E2+-+1%29 = sqrt%284%5E2-1%29 = sqrt%2815%29, according to that lesson.

So, the ellipse canonical equation is 

x%5E2%2F4%5E2+%2B+y%5E2%2F%28sqrt%2815%29%29%5E2 = 1,   or

x%5E2%2F16+%2B+y%5E2%2F15 = 1.