At what angle must the two forces (x + y) and (x - y) act so that the resultant may be √x2 + y2 ?

Question

At what angle must the two forces (x + y) and (x - y) act so that the resultant may be √x2 + y2 ? (A)  cos-1 Bigg[ - (x2 + y2/2(x2 - y2)) Bigg] (B)  cos-1 Bigg[ - (2(x2 - y2)/x2 + y2) Bigg] (C)  cos-1 Bigg[ - ((x2 + y2)/(x2 - y2)) Bigg] (D) cos-1 Bigg[ - ((x2 - y2)/(x2 + y2)) Bigg]

Tardigrade Admin · Accepted Answer

(√x2+y2)2=(x+y)2+(x-y)2+2(x+y)(x-y) cos θ x2+y2=x2+y2+2 x y+x2+y2-2 x y+2(x2-y2) cos θ 2(x2-y2) ⋅ cos θ=-(x2+y2) ⇒ θ= cos -1[(-(x2+y2)/2(x2-y2))]

Q. At what angle must the two forces (x + y) and (x - y) act so that the resultant may be $x^{2} + y^{2}$ ?

$cos^{- 1} [- \frac{x ^{2} + y ^{2}}{2 ( x ^{2} - y ^{2} )}]$

$cos^{- 1} [- \frac{2 ( x ^{2} - y ^{2} )}{x ^{2} + y ^{2}}]$

$cos^{- 1} [- \frac{( x ^{2} + y ^{2} )}{( x ^{2} - y ^{2} )}]$

$co s^{- 1} [- \frac{( x ^{2} - y ^{2} )}{( x ^{2} + y ^{2} )}]$

Q. At what angle must the two forces (x + y) and (x - y) act so that the resultant may be x2+y2​ ?

cos−1[−2(x2−y2)x2+y2​]

cos−1[−x2+y22(x2−y2)​]

cos−1[−(x2−y2)(x2+y2)​]

cos−1[−(x2+y2)(x2−y2)​]

(x2+y2​)2=(x+y)2+(x−y)2+2(x+y)(x−y)cosθ x2+y2=x2+y2+2xy+x2+y2−2xy+2(x2−y2)cosθ 2(x2−y2)⋅cosθ=−(x2+y2) ⇒θ=cos−1[2(x2−y2)−(x2+y2)​]

Q. At what angle must the two forces (x + y) and (x - y) act so that the resultant may be $x^{2} + y^{2}$ ?

$cos^{- 1} [- \frac{x ^{2} + y ^{2}}{2 ( x ^{2} - y ^{2} )}]$

$cos^{- 1} [- \frac{2 ( x ^{2} - y ^{2} )}{x ^{2} + y ^{2}}]$

$cos^{- 1} [- \frac{( x ^{2} + y ^{2} )}{( x ^{2} - y ^{2} )}]$

$co s^{- 1} [- \frac{( x ^{2} - y ^{2} )}{( x ^{2} + y ^{2} )}]$