[tex]\large\underline{\sf{Solution-}}[/tex]
Given:
[tex] \rm \longmapsto x = a \sin \alpha \cos \beta [/tex]
[tex] \rm \longmapsto y = b \sin \alpha \sin \beta [/tex]
[tex] \rm \longmapsto z = c\cos \alpha[/tex]
Therefore:
[tex] \rm \longmapsto \dfrac{x}{a} = \sin \alpha \cos \beta [/tex]
[tex] \rm \longmapsto \dfrac{y}{b} = \sin \alpha \sin \beta [/tex]
[tex] \rm \longmapsto \dfrac{z}{c} = \cos \alpha[/tex]
Now:
[tex] \rm = \dfrac{ {x}^{2} }{ {a}^{2}} + \dfrac{ {y}^{2} }{ {b}^{2} } + \dfrac{ {z}^{2} }{ {c}^{2} } [/tex]
[tex] \rm = { \sin}^{2} \alpha \cos^{2} \beta + { \sin}^{2} \alpha \sin^{2} \beta + { \cos}^{2} \alpha [/tex]
[tex] \rm = { \sin}^{2} \alpha (\cos^{2} \beta + \sin^{2} \beta )+ { \cos}^{2} \alpha [/tex]
[tex] \rm = { \sin}^{2} \alpha \cdot1+ { \cos}^{2} \alpha [/tex]
[tex] \rm = { \sin}^{2} \alpha + { \cos}^{2} \alpha [/tex]
[tex] \rm = 1[/tex]
Therefore:
[tex] \rm \longmapsto\dfrac{ {x}^{2} }{ {a}^{2}} + \dfrac{ {y}^{2} }{ {b}^{2} } + \dfrac{ {z}^{2} }{ {c}^{2} } = 1[/tex]
