Question 1045771
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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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<pre>
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 {{{6/2}}} = 3 units  (c=3);

the minor semi-axis "b" is {{{sqrt(5^2-3^2)}}} = {{{sqrt(16)}}} = 4  (b=4);   ( <--- here 5 = {{{10/2}}} )

the major semi-axis "a" is {{{sqrt(b^2 + c^2)}}} = {{{sqrt(4^2+3^2)}}} = {{{sqrt(25)}}} = 5 units long.

The equation of this ellipse is 

{{{(x-5)^2/5^2}}} + {{{(y-1)^2/4^2}}} = 1.
</pre>

See the lessons on ellipses

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-definition--canonical-equation--characteristic-points-and-elements.lesson>Ellipse definition, canonical equation, characteristic points and elements</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Ellipse-focal-property.lesson>Ellipse focal property</A>  

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangen-lines-to-a-circle.lesson>Tangent lines and normal vectors to a circle</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Tangent-lines-to-an-ellipse.lesson>Tangent lines and normal vectors to an ellipse</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-an-ellipse.lesson>Optical property of an ellipse</A> 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/Quadratic-relations-and-conic-sections/Optical-property-of-an-ellipse-revisited.lesson>Optical property of an ellipse revisited</A> 

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