Question

Find the area of the ellipse (2x + 5y − 3)
2
+ (3x − 7y + 8)
2
< 1.
I have no idea what this figure looks like.

Answer 1

Let u = 3 x - 7 y + 8, v = 2x + 5y + 3.
\[\left[\begin{matrix}u _{x} & u _{y} \\ v _{x} & v _{y}\end{matrix}\right] = \left[\begin{matrix}3& -7 \\ 2 & 5\end{matrix}\right].\]
The determinant of the Jacobian matrix is 29. The equation of the ellipse transforms to u^2 + v^2 = 1 which has an area of pi. The required area is 29 pi.