Question

prove that x² + y ² = a² + b²​

Answer 1

Answer:

Step-by-step explanation:

x=asecθ+btanθ

y=atanθ+bsecθ

x  

2

−y  

2

=a  

2

sec  

2

θ+b  

2

tan  

2

θ+2absecθtanθ−a  

2

tan  

2

θ−b  

2

sec  

2

θ−2absecθtanθ

=a  

2

(sec  

2

θ−tan  

2

θ)−b  

2

(sec  

2

θ−tan  

2

θ)

=a  

2

−b  

2

  ( As sec  

2

θ−tan  

2

θ=1)

∴x  

2

−y  

2

=a  

2

−b  

2

Answer 2

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☃ ƛƝƧƜЄƦ ☃

$( a+ i.b)^2 = x + i.y.$

$or, a^2+ i^2.b^2+ i.2ab = x + i.y.$

$or, \: a^2 - b^2 + i. 2ab = x + i. y.$

$or, Comparing \: the \ \: real \: and \: imaginary \: parts \: \: both \: sides.$

$x = a^2 - b^2. …………(1).$

$y = 2ab. …………………..(2)$

$Squaring \: \: and \: adding \: both \: sides \: the \: \: eqn. \: (1) \: \: and \: \: (2).$

$x^2 + y^2 = (a^2 - b^2)^2 +(2ab)^2.$

$or, x^2 - y^2 = a^4+b^4-2a^2.b^2+ 4a^2b^2 = a^4+b^4+2a^2b^2.$

$or, x^2 - y^2 = (a^2+b^2)^2. \\ \\ Proved.$

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