STEP III 1990 question 6 solution

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So applying the transformations to the vectors (1, 2) and (2, -1), we get (2, 4) and (8, -4), both of which are scalar multiples of the original vectors.

The matrix transforms the general point (x, y) to (X, Y) =

Therefore the function is transformed to:

As required.

The area is equal to the original area multiplied by the determinant of the matrix transformation. The original area is ; the determinant is 8. Therefore the area is .

To find the maximum value of X we find

Differentiating implicitly, we get

There is a stationary point when

Treating the curves equation as a quadratic in Y, we get

At the stationary point,

The point (0.6, 0.8) is transformed to the point .

At this point,

The equation of the tangent at this point is therefore .

Solution by Dystopia.