Question

Solve $foci\:\frac{\left(x-1\right)^2}{9}+\frac{y^2}{5}=100$

Answer 1

$\mathrm{Ellipse\:foci\:given}\:\frac{\left(x-1\right)^2}{9}+\frac{y^2}{5}=100$

$\left(h+c,\:k\right),\:\left(h-c,\:k\right)$

$Calculate\:ellipse\:properties$

$\frac{\left(x-1\right)^2}{9}+\frac{y^2}{5}=100:\quad \mathrm{Ellipse\:with\:center}\:\left(h,\:k\right)=\left(1,\:0\right),\:\mathrm{semi-major\:axis}\:a=30,\:\mathrm{semi-minor\:axis}\:b=10\sqrt{5}$

$\left(1+c,\:0\right),\:\left(1-c,\:0\right)$

$\mathrm{Compute}\:c:$

$c=\sqrt{30^2-\left(10\sqrt{5}\right)^2}$

$=20$

$\left(1+20,\:0\right),\:\left(1-20,\:0\right)$

$\mathrm{Refine}$

$\left(21,\:0\right),\:\left(-19,\:0\right)$

Answer 2

Explanation:

$\sf \to\:\frac{\left(x-1\right)^2}{9}+\frac{y^2}{5}=100 \\ \\ \sf \to \: \frac{ {x}^{2} - 2x + 1 }{9} + \frac{ {y}^{2} }{5} = 100 \\ \\ \sf \to \: \frac{ {x}^{2} - 2x + 1 + 9 {y}^{2} }{9} = 500 \\ \: \\ \sf \to \: {x}^{2} - 2x + 1 + 9 {y}^{2} = 4500 \\ \\ \sf \to \red{ {x}^{2} + {9y}^{2} - 2x - 4999}$