Finding area of a region with determinant

[Lily@pie]

Homework Statement

Find the area of the region in the plane 34x²+14xy+5y²<=4

The Attempt at a Solution

I know for this question it is best to find a standard matrix transformation, A that transform a region with a known area formula to this region bounded by an equation.

So I let the area of equation, E be 34x₁²+14x₁x₂+5x₂²<=4

x=column matrix [[x₁][x₂]]
u=column matrix [[u₁][u₂]]

and area of equation, S be bounded by
(u₁+u₂)²<=4
such that E is an image of S after some transformation T

Is this an correct approach? But I couldn't find the required transformation...

Thanks!

 

[tiny-tim]

Hi Lily@pie!

(have an leq: ≤ )

Is this an correct approach?

yes

essentially you're looking for a b c and d such that (ax + by)² + (cx + dy)² = 34x²+14xy+5y²

 

[Lily@pie]

I've found that (5x+3y)²+(2x-y)² =34x²+14xy+5y²

And I know I got to find a transformation that maps this to a circle x²+y²=4

But I'm stuck here... How should I approach it in order to find the standard matrix for this transformation...

T^T

 

[tiny-tim]

Hi Lily@pie!

Isn't it just

5 3
2 -1 ?​

 

[Lily@pie]

Huh?!? That means we just need to line them up in a matrix?
Erm... A bit lost... here...

 

[tiny-tim]

u = 5x + 3y, v = 2x - y, u² + v² = 4

 

[Lily@pie]

OOHHHH! Thanks so much! OOHHHHH! =))))