since the major axis is on the $x-$ axis, the equation of the ellipse will be of the form

$\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1$           ………. $(1)$

Where, a is the semi-major axis

The ellipse passes through points $(4,\,3)$ and $(6,\,2)$ . Hence,

$\frac{16}{a^{2}}+\frac{9}{b^{2}}=1$          ………. $(2)$

$\frac{36}{a^{2}}+\frac{4}{b^{2}}=1$          ………. $(3)$

On solving equations $(2)$ and $(3),$ we obtain $a^{2}=52$ and $b^{2}=13$

Thus, the equation of the ellipse is $\frac{x^{2}}{52}+\frac{y^{2}}{13}=1$ or $x^{2}+4 y^{2}=52$