Question

If circumcentre of an equilateral triangle inscribed in

\frac{x^{2}}{a^{2}} + \frac{y^{2}}{b^{2}} = 1,

with vertices having eccentric angles

α, β, γ

respectively is

(x_{1}, y_{1}),

then find

Σ \cos α \cos β + Σ \sin α \sin β

.

Accepted Answer

Let the parametric co-ordinates of $P : (a \cos α, b \sin α)$

$Q : (a \cos β, b \sin β)$

$R : (a \cos γ, b \sin γ)$ and given that the circumcentre of the equilateral triangle are $(x_{1}, y_{1})$

So, we know the circumcentre of equilaterals

$(x_{1}, y_{1}) = (\frac{a \cos α + a \cos β + a \cos γ}{3}, \frac{b \sin α + b \sin β + b \sin γ}{3})$

$\Rightarrow \frac{3 x_{1}}{a} = \cos α + \cos β + \cos γ . . . (i)$ and

$\Rightarrow \frac{3 y_{1}}{b} = \sin α + \sin β + \sin γ . . . (i i)$

Squaring and adding equation $(i)$ and $(i i)$ ${(\frac{3 x_{1}}{a})}^{2} + {(\frac{3 y_{1}}{b})}^{2} = {(\cos α + \cos β + \cos γ)}^{2} + {(\sin α + \sin β + \sin γ)}^{2}$

$\Rightarrow \frac{9 x_{1}^{2}}{a^{2}} + \frac{9 y_{1}^{2}}{b^{2}} = \cos^{2} α + \cos^{2} β + \cos^{2} γ + 2 \cos α \cos β + 2 \cos β \cos γ + 2 \cos γ \cos α + \sin^{2} α + \sin^{2} β + \sin^{2} γ + 2 \sin α \sin β + 2 \sin β \sin γ + 2 \sin γ \sin α$

$\frac{9 x_{1}^{2}}{a^{2}} + \frac{9 y_{1}^{2}}{b^{2}} = 3 + 2 [Σ \sin α \sin β + Σ \cos α \cos β]$

$\frac{9 x_{1}^{2}}{a^{2}} + \frac{9 y_{1}^{2}}{b^{2}} - 3 = 2 [Σ \sin α \sin β + Σ \cos α \cos β]$

$\Rightarrow Σ \sin α \sin β + Σ \cos α \cos β = \frac{9 x_{1}^{2}}{2 a^{2}} + \frac{9 y_{1}^{2}}{2 b^{2}} - \frac{3}{2}$

Exercise-4

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