Let $$a$$ and $$b$$ respectively be the semi-transverse and semi-conjugate axes of a standard hyperbola whose eccentricity satisfies the equation $$9e^2 - 18e + 5 = 0$$. If $$S(5, 0)$$ is a focus and $$5x = 9$$ is the corresponding directrix of this hyperbola, then $$a^2 - b^2$$ is equal to

We are told that the hyperbola is in its standard (centre at the origin) horizontal form, so its equation can be written as $$\dfrac{x^2}{a^2}-\dfrac{y^2}{b^2}=1.$$

For this standard form we recall three very important facts:

1. The coordinates of the foci are $$(\pm c,0).$$

2. The relation between the semi-transverse axis $$a$$, the semi-conjugate axis $$b$$ and the focal distance $$c$$ is $$c^2=a^2+b^2.$$

3. The eccentricity is defined as $$e=\dfrac{c}{a},$$ and the corresponding right-hand directrix is $$x=\dfrac{a}{e}.$$

Now we turn to the data given in the question.

First we compute the eccentricity from the quadratic condition $$9e^2-18e+5=0.$$ Dividing every term by $$9$$ gives $$e^2-2e+\dfrac{5}{9}=0.$$ Using the quadratic formula $$e=\dfrac{2\pm\sqrt{(-2)^2-4\cdot1\cdot\dfrac{5}{9}}}{2} =\dfrac{2\pm\sqrt{4-\dfrac{20}{9}}}{2} =\dfrac{2\pm\sqrt{\dfrac{16}{9}}}{2} =\dfrac{2\pm\dfrac{4}{3}}{2}.$$ This produces two numerical values: $$e_1=\dfrac{2+\dfrac{4}{3}}{2}=\dfrac{\dfrac{10}{3}}{2}=\dfrac{5}{3},\qquad e_2=\dfrac{2-\dfrac{4}{3}}{2}=\dfrac{\dfrac{2}{3}}{2}=\dfrac{1}{3}.$$ Because a hyperbola must have $$e>1,$$ we select $$e=\dfrac{5}{3}.$$

The focus supplied is $$S(5,0),$$ so the focal distance is $$c=5.$$

The directrix corresponding to this focus is given as $$5x=9,$$ which is $$x=\dfrac{9}{5}.$$ According to fact 3, for the right-hand side of the hyperbola that directrix must satisfy $$x=\dfrac{a}{e}.$$ Therefore $$\dfrac{a}{e}=\dfrac{9}{5}\quad\Longrightarrow\quad a=\dfrac{9}{5}\,e.$$

Substituting our value $$e=\dfrac{5}{3}$$ we obtain $$a=\dfrac{9}{5}\left(\dfrac{5}{3}\right)=\dfrac{9}{3}=3.$$ Hence $$a^2=3^2=9.$$

With $$a$$ known and the focal distance $$c=5,$$ we use the relation $$c^2=a^2+b^2$$ (fact 2) to find $$b^2:$$ $$25=9+b^2\quad\Longrightarrow\quad b^2=25-9=16.$$

The quantity required in the question is $$a^2-b^2:$$ $$a^2-b^2=9-16=-7.$$

Hence, the correct answer is Option A.