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Question 93
Let a, b, x, y be real numbers such that $$a^2 + b^2 = 25, x^2 + y^2 = 169$$, and $$ax + by = 65$$. If $$k = ay - bx$$, then
Solution
$$\left(ax+by\right)^2=65^2$$
$$a^2x^2\ +\ b^2y^2+\ 2abxy\ =\ 65^2$$
$$k = ay - bx$$
$$k^2\ =\ a^2y^2+b^2x^2-2abxy$$
$$(a^2 + b^2)(x^2 + y^2 )= 25* 169$$
$$a^2x^2+a^2y^2+b^2x^2+b^2y^2=\ 25\times\ 169$$
$$k^2=\ 65^2\ -\ \left(25\times\ 169\right)$$
k = 0
D is the correct answer.
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