Problems from 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 $$4$$.

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

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