Question 1

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) = ?

Question 2

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

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:

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

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