SOLUTION: what is the general equation if the locus of a point which moves so that the sum of its distances from point (2,1) and (8,1) is 10?

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Question 1045771: what is the general equation if the locus of a point which moves so that the sum of its distances from point (2,1) and (8,1) is 10?
Found 2 solutions by ikleyn, josgarithmetic:
Answer by ikleyn(52814) About Me  (Show Source):
You can put this solution on YOUR website!
.
what is the general equation if the locus of a point which moves so that the sum of its distances from point (2,1) and (8,1) is 10?
~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ 

This locus is an ellipse with the foci at the points (2,1) and (8,1);

the major axis is directed along (parallel to) x- axis; 

The center is at the point (5,1);

the focal distance is 8-2 = 6;

the linear eccentricity "c" is 6%2F2 = 3 units  (c=3);

the minor semi-axis "b" is sqrt%285%5E2-3%5E2%29 = sqrt%2816%29 = 4  (b=4);   ( <--- here 5 = 10%2F2 )

the major semi-axis "a" is sqrt%28b%5E2+%2B+c%5E2%29 = sqrt%284%5E2%2B3%5E2%29 = sqrt%2825%29 = 5 units long.

The equation of this ellipse is 

%28x-5%29%5E2%2F5%5E2 + %28y-1%29%5E2%2F4%5E2 = 1.

See the lessons on ellipses
    - Ellipse definition, canonical equation, characteristic points and elements
    - Ellipse focal property
    - Tangent lines and normal vectors to a circle
    - Tangent lines and normal vectors to an ellipse
    - Optical property of an ellipse
    - Optical property of an ellipse revisited
in this site.


Answer by josgarithmetic(39620) About Me  (Show Source):
You can put this solution on YOUR website!
The way this starts is, sqrt%28%28x-2%29%5E2%2B%28y-1%29%5E2%29%2Bsqrt%28%28x-8%29%5E2%2B%28y-1%29%5E2%29=10.

sqrt%28%28x-2%29%5E2%2B%28y-1%29%5E2%29=10-sqrt%28%28x-8%29%5E2%2B%28y-1%29%5E2%29

Square both sides.



Notice the expression %28y-1%29%5E2 shown summed on both sides.

%28x-1%29%5E2=100-20%2Asqrt%28%28x-8%29%5E2%2B%28y-1%29%5E2%29%2B%28x-8%29%5E2

%28x-1%29%5E2-100-%28x-8%29%5E2=-20%2Asqrt%28%28x-8%29%5E2%2B%28y-1%29%5E2%29

Square both sides again.
%28%28x-1%29%5E2-%28x-8%29%5E2-100%29%5E2=400%28%28x-8%29%5E2%2B%28y-1%29%5E2%29

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