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 $$4$$.
Development:
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: $$$\dfrac{(y+1)^2}{25}+\dfrac{(x-1)^2}{16}=1$$$
Solution:
a) The equation is: $$\dfrac{x^2}{9}+\dfrac{y^2}{5}=1$$
b) The equation is: $$\dfrac{(y+1)^2}{25}+\dfrac{(x-1)^2}{16}=1$$