The eccentricity of the hyperbola whose length of its conjugate axis is equal to half of the distance between its foci, is

Let us consider a standard rectangular hyperbola whose transverse axis is taken along the $$x$$-axis. Its Cartesian equation in the simplest form is written as

$$\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1.$$

For this hyperbola we know the following standard facts:

• The coordinates of the foci are $$(\pm ae,0)$$, so the distance between the two foci is $$2ae.$$

• The length of the conjugate axis (the axis along the $$y$$-direction) is twice the semi-conjugate, that is $$2b.$$

• The relation among the semi-transverse $$a$$, the semi-conjugate $$b$$ and the eccentricity $$e$$ is $$b^{2}=a^{2}(e^{2}-1).$$

Now we translate the verbal condition of the problem into an algebraic equation. We are told that “the length of its conjugate axis is equal to half of the distance between its foci.” That sentence becomes, in symbols,

$$2b=\tfrac12\,(2ae).$$

Simplifying the right-hand side first: $$\tfrac12\,(2ae)=ae.$$ Hence the given condition is

$$2b=ae.$$

We isolate $$b$$ because the formula that connects $$a,b,e$$ has $$b^{2}$$ in it. Dividing both sides by 2 gives

$$b=\frac{ae}{2}.$$

Next we square this result so that we may substitute into the basic identity: $$b^{2}=\left(\frac{ae}{2}\right)^{2}=\frac{a^{2}e^{2}}{4}.$$

But from the standard relation quoted earlier we also have $$b^{2}=a^{2}(e^{2}-1).$$

Because both right-hand sides represent the same $$b^{2},$$ we equate them:

$$\frac{a^{2}e^{2}}{4}=a^{2}(e^{2}-1).$$

Since $$a^{2}$$ is positive and common on both sides, we cancel it to obtain

$$\frac{e^{2}}{4}=e^{2}-1.$$

To clear the fraction we multiply every term by 4, giving

$$e^{2}=4e^{2}-4.$$

Now we bring all terms to one side: $$0=4e^{2}-4-e^{2}=3e^{2}-4.$$

Rearranging, we write

$$3e^{2}=4.$$

Dividing by 3 gives

$$e^{2}=\frac{4}{3}.$$

Taking the positive square root (because the eccentricity of a hyperbola is always >1) yields

$$e=\frac{2}{\sqrt{3}}.$$

We compare this with the answer choices and see that it matches Option A.

Hence, the correct answer is Option A.