Answer to Question #160250 in Linear Algebra for Nikhil Singh

Question #160250

Reduce the quadratic form 5x^2-4xy+8y^2 to its orthogonal canonical form, clearly giving the transformations being used. Also give a rough sketch of the curve representing this canonical form.



1
Expert's answer
2021-02-24T07:07:02-0500

5x24xy+8y2=15x^2 -4xy +8y^2 = 1 (instead of one, any number can be here, you did not specify it)

(nx±my)2=c(nx \pm my)^2 = c - canonical form

first of all we should find the square roots of the first and last terms:

5x2=5x\sqrt{5x^2} = \sqrt{5}*x

8y2=22y\sqrt{8y^2} = 2\sqrt{2}*y


-4 - should be multiplication of 5\sqrt{5} and 222\sqrt{2} and + free term:


522=210\sqrt{5}*2\sqrt{2} = 2\sqrt{10}


Answer: (5x+22y)2(4+210)xy=1(\sqrt{5}x + 2\sqrt{2}y)^2 - (4+2\sqrt{10})xy = 1



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