- #1

utkarshakash

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## Homework Statement

The equation [itex]ax^2 + 2hxy+by^2=1 [/itex] represents the equation of ellipse if [itex]h^2-ab<0[/itex]. When x=Xcosθ-Ysiinθ and y=Xsinθ+Ycosθ, the above equation transforms to

[itex]\dfrac{X^2}{\alpha^2} + \dfrac{Y^2}{\beta ^2} = 1 [/itex] where σ and β are real numbers. Then find the product of major and minor axis of this ellipse in terms of h,a,b.

## The Attempt at a Solution

I tried transforming the original equation and ended up with this:

[itex](a \cos ^2 \theta + b \sin ^2 \theta + 2h \cos \theta \sin \theta)X^2[/itex]

[itex] + (a \sin^2 \theta + b \cos ^2 \theta -2h \sin \theta \cos \theta) Y^2[/itex]

[itex] + (-2a \cos \theta \sin \theta +2b \sin \theta \cos \theta +2h \cos^2 \theta - 2h \sin ^2 \theta) = 1 [/itex]

The product of the major and minor axis will be 4αβ. For finding αβ I tried to find the product of coefficients of X^2 and Y^2. But the expression seems too complicated.

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