Solve (ax-by)^{2}-(ay-bx)^{2}-(a^2-b^2)(x^2-y^2)

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

Evaluate

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Solution Steps

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(ax-by\right)^{2}.

$$a^{2}x^{2}-2axby+b^{2}y^{2}-\left(ay-bx\right)^{2}-\left(a^{2}-b^{2}\right)\left(x^{2}-y^{2}\right)$$

Use binomial theorem \left(p-q\right)^{2}=p^{2}-2pq+q^{2} to expand \left(ay-bx\right)^{2}.

$$a^{2}x^{2}-2axby+b^{2}y^{2}-\left(a^{2}y^{2}-2aybx+b^{2}x^{2}\right)-\left(a^{2}-b^{2}\right)\left(x^{2}-y^{2}\right)$$

To find the opposite of a^{2}y^{2}-2aybx+b^{2}x^{2}, find the opposite of each term.

$$a^{2}x^{2}-2axby+b^{2}y^{2}-a^{2}y^{2}+2aybx-b^{2}x^{2}-\left(a^{2}-b^{2}\right)\left(x^{2}-y^{2}\right)$$

Combine -2axby and 2aybx to get 0.

$$a^{2}x^{2}+b^{2}y^{2}-a^{2}y^{2}-b^{2}x^{2}-\left(a^{2}-b^{2}\right)\left(x^{2}-y^{2}\right)$$

Use the distributive property to multiply a^{2}-b^{2} by x^{2}-y^{2}.

$$a^{2}x^{2}+b^{2}y^{2}-a^{2}y^{2}-b^{2}x^{2}-\left(a^{2}x^{2}-a^{2}y^{2}-b^{2}x^{2}+y^{2}b^{2}\right)$$

Combine a^{2}x^{2}+b^{2}y^{2}-a^{2}y^{2}-b^{2}x^{2} and -\left(a^{2}x^{2}-a^{2}y^{2}-b^{2}x^{2}+y^{2}b^{2}\right) to get 0.

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Factor

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