First slide

Cartesian plane

Question

If the distance between two points $(a \cos θ, a \sin θ)$ and $(a \cos ϕ, a \sin ϕ)$ is 2a then $θ$ is

Moderate

A

$2 n π + π + ϕ, n \in z$
B

$n π + \frac{π}{2} + ϕ, n \in z$
C

$n π - ϕ, n \in z$
D

$2 n π + ϕ, n \in z$

Solution

${(a \cos θ - a \cos ϕ)}^{2} + {(a \sin θ - a \sin ϕ)}^{2} = 4 a^{2}$
$\Rightarrow - 2 a^{2} \cos (θ - ϕ) = 4 a^{2} - 2 a^{2}$
$\Rightarrow \cos (θ - ϕ) = - 1$
$\Rightarrow θ - ϕ = 2 n π \pm π$
$\Rightarrow θ = 2 n π \pm π + ϕ, n \in Z$

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