If a 2+ b 2=1, m 2+ n 2=1, then which of the following is true for all values of m , n , a , b -

Question

If a 2+ b 2=1, m 2+ n 2=1, then which of the following is true for all values of m , n , a , b - (A) |a m+b n| ≤ 1 (B)  mid am - bn mid ≥ 1 (C) | am + bn | ≥ 1 (D)  mid am - bn mid ≤ 1

Tardigrade Admin · Accepted Answer

Consider a = cos θ, b = sin θ m = cos φ, n = sin φ Now, am ± bn = cos θ cos φ ± sin θ sin φ am ± bn = cos (θ ∓ φ) ∴| am ± bn | ≤ 1

Q. If $a^{2} + b^{2} = 1, m^{2} + n^{2} = 1$ , then which of the following is true for all values of $m, n, a, b$ -

$∣ am + bn ∣ \leq 1$

$∣$ am $-$ bn $∣\geq 1$

$∣ am + bn ∣ \geq 1$

$∣$ am $-$ bn $∣\leq 1$

Consider $a = cos θ, b = sin θ$
$m = cos ϕ, n = sin ϕ$
Now, am $\pm bn = cos θ cos ϕ \pm sin θ sin ϕ$
$am \pm bn = cos (θ \mp ϕ)$
$∴ ∣ am \pm bn ∣ \leq 1$

Q. If a2+b2=1,m2+n2=1, then which of the following is true for all values of m,n,a,b -

∣am+bn∣≤1

∣ am − bn ∣≥1

∣am+bn∣≥1

∣ am − bn ∣≤1