cat 2019 Question 27

Question

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

Accepted Answer

$\left(ax+by ight)^2=65^2$

$a^2x^2\ +\ b^2y^2+\ 2abxy\ =\ 65^2$

$k = ay - bx$

$k^2\ =\ a^2y^2+b^2x^2-2abxy$

$(a^2 + b^2)(x^2 + y^2 )= 25* 169$

$a^2x^2+a^2y^2+b^2x^2+b^2y^2=\ 25 imes 169$

$k^2=\ 65^2\ -\ \left(25 imes\ 169 ight)$

k = 0

D is the correct answer.

Question 27.