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

The equation of the ellipse is

`((x-2)^(2) )/(100) +((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=√(64)\\ c=8`

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

The ladus rectus is

`LR=(2b^(2) )/(a)=(2(36))/(10)=7.2`

The eccentricity is

`e=(c)/(a)=(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)