What is (ax+by)^2+(ay-bx)^2 ?

The solution to (ax+by)^2+(ay-bx)^2 is a^2x^2+b^2y^2+a^2y^2+b^2x^2

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simplify (ax+by)^2+(ay-bx)^2

simplify

(a x + b y)^{2} + (a y - b x)^{2}

a^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2} + b^{2} x^{2}

Solution steps

(a x + b y)^{2} + (a y - b x)^{2}

Expand

(a x + b y)^{2} : a^{2} x^{2} + 2 ab x y + b^{2} y^{2}

= a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + (a y - b x)^{2}

Expand

(a y - b x)^{2} : a^{2} y^{2} - 2 ab x y + b^{2} x^{2}

= a^{2} x^{2} + 2 ab x y + b^{2} y^{2} + a^{2} y^{2} - 2 ab x y + b^{2} x^{2}

Group like terms

= a^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2} + b^{2} x^{2} + 2 ab x y - 2 ab x y

Add similar elements:

2 ab x y - 2 ab x y = 0

= a^{2} x^{2} + b^{2} y^{2} + a^{2} y^{2} + b^{2} x^{2}

5 x - 4 y + 2 (y + x)

15 \cdot 20

\frac{[ x ^{6} ]}{4}

sin^{2} (a) + cos^{2} (a)

sin (2 t) cos (2 t)

What is (ax+by)^2+(ay-bx)^2 ?
The solution to (ax+by)^2+(ay-bx)^2 is a^2x^2+b^2y^2+a^2y^2+b^2x^2