Question

suppose (-8,0) and (8,0) are the foci of an ellipse. using the distance formula write an equation stating that the sum of the two distances from a point p(x,y) on the ellipse to the two foci is 20

Answer 1

\[\sqrt{(x+8)^2+(y-0)^2}+\sqrt{(x-8)^2+(y-0)^2}=20\]

Answer 2

now eliminate the radicals in this equation by squaring twice. simplify the resulting equation and write it in standard form of the equation of an ellipse. what are a and b for this ellipse?

Answer 3

Did you do it?

Answer 4

\[\sqrt{(x+8)^2+y^2}=20-\sqrt{(x-8)^2+y^2}\]
\[(x+8)^2+y^2=400-40\sqrt{(x-8)^2+y^2}+(x-8)^2+y^2\]
\[x^2+16x+64+y^2=400-40\sqrt{(x-8)^2+y^2}+x^2-16x+64+y^2\]
\[32x-400=-40\sqrt{(x-8)^2+y^2}\]
\[4x-50=-5\sqrt{(x-8)^2=y^2}\]
\[16x^2-400x+2500=25[(x-8)^2+y^2]\]
\[16x^2-400x+2500=25(x^2-16x+64+y^2)\]
\[16x^2-400x+2500=25x^2-400x+1600+25y^2\]
\[900=9x^2+25y^2\]

Answer 5

Divide both sides by 900

Answer 6

\[\frac{x^2}{100}+\frac{y^2}{36}=1\]

Answer 7

a=10, b=6