if x = a cos theta and y = b sin theta , then b^2 x^2+a^2 y^2 =
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Work 1:
Given, x = a cos theta and y = b sin theta. Substituting for x and y ,
b²x² + a²y² - a²b² = b²a² cos² theta + a²b² sin² theta - a²b²
= a²b² (cos² theta + sin² theta) - a²b²
= a²b² . 1 - a²b² (Using the trigonometric identity cos² theta + sin² theta = 1)
= a²b² - a²b² = 0 (Proved)
Work 2:
x = a cos theta , y = b sin theta . Substituting for x,
b²x² + a²y² - a²b² = b²a² cos² theta - a²b² + a²y²
= b²a²(cos² theta-1) + a²y²
= b²a²(-sin² theta) + a²y² = -a²(b²sin² theta) + a²y² Substitute y for b sin theta.
=-a²y² + a²y²
= 0 (Proved)
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