Question

If x = a sin α, cos β, y = b sin α.sin β and z = c cos α then (x²/a²) + (y²/b²) + (z²/c²) = ?​

Answer 1

`\large\underline{\sf{Solution-}}`

Given:

`\rm \longmapsto x = a \sin \alpha \cos \beta`

`\rm \longmapsto y = b \sin \alpha \sin \beta`

`\rm \longmapsto z = c\cos \alpha`

Therefore:

`\rm \longmapsto (x)/(a) = \sin \alpha \cos \beta`

`\rm \longmapsto (y)/(b) = \sin \alpha \sin \beta`

`\rm \longmapsto (z)/(c) = \cos \alpha`

Now:

`\rm = \frac{ {x}^(2) }{ {a}^(2)} + \frac{ {y}^(2) }{ {b}^(2) } + \frac{ {z}^(2) }{ {c}^(2) }`

`\rm = { \sin}^(2) \alpha \cos^(2) \beta + { \sin}^(2) \alpha \sin^(2) \beta + { \cos}^(2) \alpha`

`\rm = { \sin}^(2) \alpha (\cos^(2) \beta + \sin^(2) \beta )+ { \cos}^(2) \alpha`

`\rm = { \sin}^(2) \alpha \cdot1+ { \cos}^(2) \alpha`

`\rm = { \sin}^(2) \alpha + { \cos}^(2) \alpha`

`\rm = 1`

Therefore:

`\rm \longmapsto\frac{ {x}^(2) }{ {a}^(2)} + \frac{ {y}^(2) }{ {b}^(2) } + \frac{ {z}^(2) }{ {c}^(2) } = 1`