Let k be a real number such that k ≠ 0. If α and β are non zero complex numbers satisfying α+β=-2 k and α2+β2=4 k2-2 k, then a quadratic equation having (α+β/α) and (α+β/β) as its roots is equal to

Question

Let k be a real number such that k ≠ 0. If α and β are non zero complex numbers satisfying α+β=-2 k and α2+β2=4 k2-2 k, then a quadratic equation having (α+β/α) and (α+β/β) as its roots is equal to (A) 4 x2-4 k x+k=0 (B) x 2-4 kx +4 k =0 (C) 4 kx 2-4 x + k =0 (D) 4 k x2-4 k x+1=0

Tardigrade Admin · Accepted Answer

We have α+β=-2 k Also, α2+β2=4 k 2-2 k ⇒(α+β)2-2 α β=4 k 2-2 k ⇒ α β= k ∴ Sum of roots =(α+β/α)+(α+β/β) =((α+β)2/α β)=(4 k 2/ k )=4 k and product of roots =((α+β/α))((α+β/β)) =(4 k 2/ k )=4 k Hence required quadratic equation is x 2-4 kx +4 k =0

Q. Let $k$ be a real number such that $k \neq = 0$ . If $α$ and $β$ are non zero complex numbers satisfying $α + β = - 2 k$ and $α^{2} + β^{2} = 4 k^{2} - 2 k$ , then a quadratic equation having $\frac{α + β}{α}$ and $\frac{α + β}{β}$ as its roots is equal to

$4 x^{2} - 4 k x + k = 0$

$x^{2} - 4 k x + 4 k = 0$

$4 k x^{2} - 4 x + k = 0$

$4 k x^{2} - 4 k x + 1 = 0$

Q. Let k be a real number such that k=0. If α and β are non zero complex numbers satisfying α+β=−2k and α2+β2=4k2−2k, then a quadratic equation having αα+β​ and βα+β​ as its roots is equal to

4x2−4kx+k=0

x2−4kx+4k=0

4kx2−4x+k=0

4kx2−4kx+1=0

We have α+β=−2k Also, α2+β2=4k2−2k ⇒(α+β)2−2αβ=4k2−2k ⇒αβ=k ∴ Sum of roots =αα+β​+βα+β​ =αβ(α+β)2​=k4k2​=4k and product of roots =(αα+β​)(βα+β​) =k4k2​=4k Hence required quadratic equation is x2−4kx+4k=0

Q. Let $k$ be a real number such that $k \neq = 0$ . If $α$ and $β$ are non zero complex numbers satisfying $α + β = - 2 k$ and $α^{2} + β^{2} = 4 k^{2} - 2 k$ , then a quadratic equation having $\frac{α + β}{α}$ and $\frac{α + β}{β}$ as its roots is equal to

$4 x^{2} - 4 k x + k = 0$

$x^{2} - 4 k x + 4 k = 0$

$4 k x^{2} - 4 x + k = 0$

$4 k x^{2} - 4 k x + 1 = 0$