If x=a sec A cos B, y=b sec A sin B and z=c tan A, then

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If

x = a sec A cos B

,

y = b sec A sin B

and

z = c tan A

, then

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} =

Options:

A . 0

B . 3

C . 2

D . 1

Answer: Option D
:
D

x = a sec A cos B \Rightarrow \frac{x}{a} = sec A cos B

y = b sec A sin B \Rightarrow \frac{y}{b} = sec A sin B

z = c tan A \Rightarrow \frac{z}{c} = tan A

Squaring each of the equations we get,

\frac{x^{2}}{a^{2}} = {sec}^{2} A {cos}^{2} B, \frac{y^{2}}{b^{2}} = {sec}^{2} A {sin}^{2} B, \frac{z^{2}}{c^{2}} = {tan}^{2} A

∴ \frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} - \frac{z^{2}}{c^{2}} = {sec}^{2} A {cos}^{2} B + {sec}^{2} A {sin}^{2} B - {tan}^{2} A

= {sec}^{2} A ({cos}^{2} B + {sin}^{2} B) - {tan}^{2} A

= {sec}^{2} A - {tan}^{2} A \dots \dots ({cos}^{2} B + {sin}^{2} B = 1)

= 1 \dots \dots ({sec}^{2} A - {tan}^{2} A = 1)

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