Question

Una elipse se describe segun la ecuacion (x-2)2 +(y-1)2 halla las coordenadas de los vertices, focos, las longitudes de las ejes mayor y menor, el valor de la excentricidad, la longitud de los lados rectos y realiza la representacion grafica

Answer 1

Answer:

The equation of the ellipse is

$\frac{(x-2)^{2} }{100} +\frac{(y-1)^{2} }{36} =1$

This is the explicit form of an ellipse, where $h=2$ and $k=1$, which means the center of the ellipse is at $C(2,1)$.

Remember that the greatest denominator in an ellipse is the parameter $a^{2}$ and the least is $b^{2}$, so

$a^{2}=100 \implies a=10\\ b^{2}=36 \implies b=6$

The length of the major axis is: $2a=2(10)=20$

The length of the minor axis is: $2b=2(6)=12$

Vertices are $(-8,1)$ and $(12,1)$, because the center is not the origin of the coordinate system so, the vertices are displaced. The least axis vertices are: $(2,7)$ and $(2,-5)$.

The foci are on the major axis, so their vertical coordinate is 1, their horizontal coordinate depends on parameter $c$, which is related as the following

$a^{2}=b^{2} +c^{2} \\100-36=c^{2}\\ c=\sqrt{64}\\ c=8$

Therefore, the foci coordinates are $F(10,1)$ and $F'(-6,1)$.

The ladus rectus is

$LR=\frac{2b^{2} }{a}=\frac{2(36)}{10}=7.2$

The eccentricity is

$e=\frac{c}{a}=\frac{8}{10}=0.8$

Additionally, the graph is attached, there you can observe some elements of the ellipse.

(Remember, all elements of an ellipse depend on the three parameters a, b and c, that's what you need to find first)