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Step 1 :

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Rotation formula :

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If the x and y-axes are rotated through an angle $\"\"$ , the coordinates $\"\"$ of a point P relative to the xy -plane and the coordinates $\"\"$ of the same point relative to the new x and y-axis and are related by the formulas $\"\"$ and $\"\"$ .

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The general form is $\"\"$

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The angle is $\"\"$ .

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Step 2 :

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The equation is $\"\"$ .

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Compare $\"\"$ with $\"\"$

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$\"\"$ and $\"\"$ .

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The angle is $\"\"$ .

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Substitute $\"\"$ and $\"\"$ in $\"\"$

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$\"\"$

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Step 3 :

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Rotation of x - axis :

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$\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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Rotation of y - axis :

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$\"\"$ .

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Substitute $\"\"$ in above equation.

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$\"\"$

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The rotation formulas are $\"\"$ and $\"\"$ ..

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Step 4 :

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Substitute $\"\"$ and $\"\"$ in $\"\"$ .

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$\"\"$

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Divide each side 12.

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$\"\"$

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The above equation is an ellipse.

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Step 6 :

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The graph of the function $\"\"$ .

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$\"\"$

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Solution :

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The equation of an ellipse is $\"\"$

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The graph of the function $\"\"$ .

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$\"\"$

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