If-X-2-Y-2-10-XY-5-Find-X-2-Y-2-

Question Number 86085 by Serlea last updated on 27/Mar/20

If X^{2} + Y^{2} = 10

XY = 5

Find (X^{2} - Y^{2})

Commented by jagoll last updated on 27/Mar/20

{(x + y)}^{2} = 10 + 2 xy

x + y = \pm \sqrt{20} = \pm 2 \sqrt{5}

{(x - y)}^{2} = {(x + y)}^{2} - 4 xy

{(x - y)}^{2} = {(2 \sqrt{5})}^{2} - 4.5

\Rightarrow {(x - y)}^{2} = 20 - 20 = 0

\Rightarrow x = y

x^{2} - y^{2} = 0

Commented by Prithwish Sen 1 last updated on 27/Mar/20

b u t, s i r

{(x - y)}^{2} = x^{2} + y^{2} - 2 x y = 10 - 2.5 = 0

Commented by jagoll last updated on 27/Mar/20

yes . sorry . made mistake

Commented by Prithwish Sen 1 last updated on 27/Mar/20

{It}^{'} s ok sir . Have a nice day .

Answered by mr W last updated on 27/Mar/20

X^{2} + Y^{2} = 10

X^{2} Y^{2} = 25

z^{2} - 10 z + 25 = 0

{(z - 5)}^{2} = 0

z = 5

\Rightarrow X^{2} = Y^{2} = 5

\Rightarrow X^{2} - Y^{2} = 0

Commented by Prithwish Sen 1 last updated on 27/Mar/20

e x c e l l e n t s i r .

Commented by Serlea last updated on 27/Mar/20

Ok

X^{2} + Y^{2} = 10

xy = 5

2 (xy) = 2 (5) - 2 (xy) = - 2 (5)

2 xy = 10 - 2 xy = - 10

x^{2} + 2 xy + y^{2} = 10 + 10 x^{2} - 2 xy + y^{2} = 10 - 10

{(x + y)}^{2} = 20 {(x - y)}^{2} = 0

x + y = \sqrt{20} x - y = 0

Therefore

x^{2} - Y^{2} = (x - y) (x + y) = 0

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