Equation of the ellipse with center (x0, y0) and focal axis parallel to y axis

Find the equation of the ellipse knowing that it is:

a) Centred on the origin with focus $(2, 0)$ and with major semiaxis measuring $3$.

b) Centred in $(1,-1)$ with focus $(1, 2)$ and minor semiaxis $4$.

a) For this case, since it is centred on the zero and the focus is in the axis $OX$, we use the first equation of the ellipse.

We see that the major semiaxis measures $3$ and the equation is: $$\dfrac{x^2}{3^2}+\dfrac{y^2}{b^2}=1 \Rightarrow \text{ since } c=2 \text{ we obtain } \ b^2=3^2-2^2=5 \Rightarrow b=\sqrt{5}$$

The equation will be $\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$.

b) For this case, since it is not centred on the zero and given that it has the focus in the axis that is parallel to the $OY$, we use the formula. We also know that $b=4$ and $c=3$ therefore $a$ is: $a=\sqrt{16+9}=5$. Then the equation is: $$\dfrac{(y+1)^2}{25}+\dfrac{(x-1)^2}{16}=1$$

a) The equation is: $\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$

b) The equation is: $\dfrac{(y+1)^2}{25}+\dfrac{(x-1)^2}{16}=1$

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