SOLUTION: a set of points P(x,y) has the property that the distance from (-2,0) plust the ditance from (4,0) is 8. Derive the equation of the conic (using distances) and verify that it is an

Algebra ->  Quadratic-relations-and-conic-sections -> SOLUTION: a set of points P(x,y) has the property that the distance from (-2,0) plust the ditance from (4,0) is 8. Derive the equation of the conic (using distances) and verify that it is an      Log On


   

Question 838920: a set of points P(x,y) has the property that the distance from (-2,0) plust the ditance from (4,0) is 8. Derive the equation of the conic (using distances) and verify that it is an ellipse.
Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
Sum of two distances must be 8 units.

Distance from P to (-2,0) PLUS Distance from P to (4,0) EQUALS 8.

You use the Distance Formula.

sqrt%28%28x-%28-2%29%29%5E2%2B%28y-0%29%5E2%29%2Bsqrt%28%28x-4%29%5E2%2B%28y-0%29%5E2%29=8
sqrt%28%28x%2B2%29%5E2%2By%5E2%29%2Bsqrt%28%28x-4%29%5E2%2By%5E2%29=8
sqrt%28%28x%2B2%29%5E2%2By%5E2%29=8-sqrt%28%28x-4%29%5E2%2By%5E2%29
Square both sides;
%28x%2B2%29%5E2%2By%5E2=64-16%2Asqrt%28%28x-4%29%5E2%2By%5E2%29%2B%28x-4%29%5E2%2By%5E2
%28x%2B2%29%5E2=-16%2Asqrt%28%28x-4%29%5E2%2By%5E2%29%2B%28x-4%29%5E2%2B64
%28x%2B2%29%5E2-%28x-4%29%5E2-64=-16%2Asqrt%28%28x-4%29%5E2%2By%5E2%29
x%5E2%2B4x%2B4-%28x%5E2-8x%2B16%29-64=-16%2Asqrt%28%28x-4%29%5E2%2By%5E2%29
---steps done on papter---
256y%5E2%2B112x%5E2-1136x=1680, and need to complete the square process for terms with x, not showing the process here...
256y%5E2%2B112%28x-71%2F14%29%5E2=1680
.
.

This is the standard form of an ellipse.
a%5E2=105%2F7 and b%5E2=105%2F16.
The center is not translated along the y axis, but is translated along the x axis, 71%2F14 units to the right.