Step 1: Analyze the Ellipse

The given equation of the ellipse is $x^2 + 4y^2 = 4$. To find its properties, we first convert it to the standard form ${\frac{x^2}{a_e^2}} + {\frac{y^2}{b_e^2}} = 1$. Dividing the equation by 4, we get: ${\frac{x^2}{4}} + {\frac{y^2}{1}} = 1$ Comparing this with the standard form, we have: $a_e^2 = 4 \implies a_e = 2$ $b_e^2 = 1 \implies b_e = 1$

The eccentricity of the ellipse, $e_e$, is given by the formula $e_e = \sqrt{1 - \frac{b_e^2}{a_e^2}}$. $e_e = \sqrt{1 - \frac{1}{4}} = \sqrt{\frac{3}{4}} = \frac{\sqrt{3}}{2}$ The foci of the ellipse are at $(\pm c_e, 0)$, where $c_e = a_e e_e$. $c_e = 2 \times \frac{\sqrt{3}}{2} = \sqrt{3}$ So, the foci of the ellipse are $(\pm\sqrt{3}, 0)$.

Step 2: Determine the Properties of the Hyperbola

The equation of the hyperbola is given as ${\frac{x^2}{a^2}} - {\frac{y^2}{b^2}} = 1$.

1. Eccentricity of the Hyperbola ($e_h$): It is given that the eccentricity of the hyperbola is the reciprocal of the eccentricity of the ellipse. $e_h = \frac{1}{e_e} = \frac{1}{\sqrt{3}/2} = \frac{2}{\sqrt{3}}$

2. Finding $a^2$ and $b^2$ for the Hyperbola: The hyperbola passes through a focus of the ellipse. Let's use the focus $(\sqrt{3}, 0)$. Since this point lies on the hyperbola, it must satisfy its equation: ${\frac{(\sqrt{3})^2}{a^2}} - {\frac{0^2}{b^2}} = 1$ ${\frac{3}{a^2}} = 1 \implies a^2 = 3$ For a hyperbola, the relationship between $a, b,$ and $e_h$ is $b^2 = a^2(e_h^2 - 1)$. We have $a^2 = 3$ and $e_h = \frac{2}{\sqrt{3}}$, so $e_h^2 = \frac{4}{3}$. $b^2 = 3\left(\frac{4}{3} - 1\right) = 3\left(\frac{1}{3}\right) = 1$

Step 3: Write the Equation and Find the Foci of the Hyperbola

With $a^2 = 3$ and $b^2 = 1$, the equation of the hyperbola is: ${\frac{x^2}{3}} - {\frac{y^2}{1}} = 1$ Multiplying by 3, we can write this as: $x^2 - 3y^2 = 3$ The foci of the hyperbola are at $(\pm c_h, 0)$, where $c_h^2 = a^2 + b^2$. $c_h^2 = 3 + 1 = 4 \implies c_h = 2$ So, the foci of the hyperbola are $(\pm 2, 0)$.

Step 4: Evaluate the Given Options

• A: the equation of the hyperbola is ${{{x^2}} \over 3} - {{{y^2}} \over 2} = 1$ Our derived equation is ${{{x^2}} \over 3} - {{{y^2}} \over 1} = 1$. This option is incorrect as $b^2=2$ is given, but we found $b^2=1$.

• B: a focus of the hyperbola is $(2, 0)$ We found the foci of the hyperbola to be $(\pm 2, 0)$. So, $(2,0)$ is a focus. This option is correct.

• C: the eccentricity of the hyperbola is $\sqrt {{5 \over 3}}$ We calculated the eccentricity to be $e_h = \frac{2}{\sqrt{3}} = \sqrt{\frac{4}{3}}$. The value given is $\sqrt{\frac{5}{3}}$. This option is incorrect.

• D: The equation of the hyperbola is ${x^2} - 3{y^2} = 3$ Our derived equation is ${{{x^2}} \over 3} - {{{y^2}} \over 1} = 1$, which is equivalent to $x^2 - 3y^2 = 3$. This option is correct.

Therefore, the correct options are B and D.