Answer to Question #255554 in Analytic Geometry for Claris

Question

a certain celestial body is orbiting around a star somewhere in space and astronaut tries to graph out the elliptical orbit of the celestial body using dimensional analysis and scaling in his drawing the celestial body is in origin and he finds out that equation of the elliptical orbit is 4x²+9y²-256x+342y+k=0 for some value of k if is finding are true find the location of the star in the cartesian plane

Answer 1

$4x^2+9y^2-256x+342y+k=(4x^2-256x)+(9y^2+342y)+k=4(x^2-64x)+9(y^2+38y)+k=4(x^2-64x+1024-1024)+9(y^2+38y+361-361)+k=4(x-32)^2-4096+9(y+19)^2-3249+k=4(x-32)^2+9(y+19)^2-7345+k=0$

$4(x-32)^2+9(y+19)^2=7345-k$

$\frac{4(x-32)^2}{7345-k}+\frac{9(y+19)^2}{7345-k}=1$

Answer: C(32,-19)