Question

If xcosA/a+ ysinA/b=1 and xsinA/a- ycosA/b=-1
prove that x^{​​​​​​2}/a^{​​​​​​2}+y^{​​​​​​2}/ b ²=2
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Answer 1

(x/a cosA + y/b sinA)² = 1
square both sides
x²/a² cos²A + y²/b² sin²A + 2xy/(ab) sinA cos A = 1 equation (1)


x/a sinA - y/b cosA = 1

square both sides
(x/a sinA - y/b cosA)² = 1
x²/a² sin²A + y²/b² cos²A - 2xy/(ab) sinA cos A = 1 equation (2)


Now add (1) and (2) :
x²/a² cos²A + y²/b² sin²A + 2xy/(ab) sinA cos A + x²/a² sin²A + y²/b² cos²A - 2xy/(ab) sinA cos A = 2
x²/a² (cos²A + sin²A) + y²/b² (sin²A + cos²A) = 2
=> x²/a² + y²/b² = 2, proved
Since sin²A + cos²A = 1