How to solve this : For what values of a , the major axis of $\frac{x^2}{\log_{1/3}a^2}+\frac{y^2}{\log_a9 -5}=1$ is x axis?

Question

For what values of $a$ , the major axis of $\frac{x^2}{\log_{1/3}a^2}+\frac{y^2}{\log_a9 -5}=1$ is x axis ?

How to solve such problems, I have no idea about this , please guide will be of great help.. Thanks

Answer 1

For the major axis of the ellipse to be the $\,x\,$-axis, we need $\,\log_{1/3}a^2 \gt \log_a9 - 5 \gt 0.\,$ First, some manipulations:

$$\log_{1/3}a^2 = 2\log_{1/3}a = -2\log_{3}a$$

$$\log_a9 = \log_a3^2 = 2\log_a3 = \dfrac{2}{\log_3a}$$

Note that

$$\log_{1/3}a^2 \gt 0 \Rightarrow -2\log_{3}a \gt 0 \Rightarrow \log_{3}a \lt 0$$

but

$$\log_a9 - 5 \gt 0 \Rightarrow \dfrac{2}{\log_3a} \gt 0 \Rightarrow \log_{3}a \gt 0$$

which is a contradiction. So the major axis cannot be the $\,x\,$-axis.