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प्रश्न
If x = a sec θ cos ϕ, y = b sec θ sin ϕ and z = c tan θ, then\[\frac{x^2}{a^2} + \frac{y^2}{b^2}\]
विकल्प
\[\frac{z^2}{c^2}\]
\[1 - \frac{z^2}{c^2}\]
\[\frac{z^2}{c^2} - 1\]
\[1 + \frac{z^2}{c^2}\]
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उत्तर
Given:
`x= a secθcosΦ`
`⇒ x/a=secθ cosΦ `
`y=b sec θ sinΦ `
`⇒ y/b=secθ sinΦ `
`z=c tan θ`
`z/c= tan θ`
Now,
`(x/a)^2+(y/b)^2-(z/c)^2=(secθ cosΦ)^2+(secθ sin Φ)^2-(tanθ )^2`
`⇒ x^2/a^2+y^2/b^2-z^2/c^2= sec^2θcos^2 Φ+sec^2θsin^2Φ-tan^2θ`
`⇒ x^2/a^2+y^2/b^2-z^2/c^2=(sec^2θ cos^2Φ+sec^2θ sin^2 sin^2Φ)-tan^2Φ`
`⇒ x^2/a^2+y^2/b^2-z^2/c^2=sec^2θ(cos^2Φ+sin^2Φ)-tan^2θ`
`⇒ x^2/a^2+y^2/b^2-z^2/c^2= sec^2θ(1)-tan^2θ`
`⇒ x^2/a^2+y^2/b^2-z^2/c^2=sec^2θ-tan^2θ`
`⇒ x^2/a^2+y^2/b^2-z^2/c^2=1`
`⇒x^2/a^2+y^2/b^2=1+z^2/c^2`
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