Question-68110

Question Number 68110 by TawaTawa last updated on 05/Sep/19

Commented by kaivan.ahmadi last updated on 07/Sep/19

{(\frac{y}{2})}^{2} = 1 - {(\frac{x}{3})}^{2} \Rightarrow y = \pm 2 \sqrt{1 - {(\frac{x}{3})}^{2}}

e q u a t i o n o f B C :

m = t g 30 = \frac{\sqrt{3}}{2}

y - 0 = \frac{\sqrt{3}}{2} (x + 3) \Rightarrow y = \frac{\sqrt{3}}{2} x + \frac{3 \sqrt{3}}{2}

e q u a t i o n o f O D :

m = t g 60 = \sqrt{3}

y = \sqrt{3} x

w e f i n d p o i n t C

{(\frac{x}{3})}^{2} + {(\frac{\sqrt{3}}{4} x + \frac{3 \sqrt{3}}{4})}^{2} = 1 \Rightarrow \frac{1}{9} x^{2} + \frac{3}{16} x^{2} + \frac{9}{8} x + \frac{27}{16} = 1 \Rightarrow

16 x^{2} + 27 x^{2} + 162 x + 243 = 144 \Rightarrow 43 x^{2} + 162 x + 99 = 0

x_{1, 2} = \frac{- 162 \pm 96}{86} = {\begin{cases} \frac{- 162 + 96}{86} = \frac{- 66}{86} = \frac{- 33}{43} \\ \frac{- 162 - 96}{86} = - 3 \end{cases}

s o C (\frac{- 33}{43}, \frac{- 33 \sqrt{3}}{86} + \frac{3 \sqrt{3}}{2}) = (\frac{- 33}{43}, \frac{96 \sqrt{3}}{86}) = (\frac{- 33}{43}, \frac{48 \sqrt{3}}{43})

s o t h i s d i a g r a m i s n o t t r u e .

a n d n o w w e f i n d p o i n t D :

{(\frac{x}{3})}^{2} + {(\frac{\sqrt{3}}{2} x)}^{2} = 1 \Rightarrow \frac{1}{9} x^{2} + \frac{3}{4} x^{2} = 1 \Rightarrow 13 x^{2} = 36 \Rightarrow

x^{2} = \frac{36}{13} \Rightarrow x = \pm \sqrt{\frac{36}{13}} \Rightarrow D (\sqrt{\frac{36}{13}}, \sqrt{\frac{108}{13}})

s o w e h a v e

S_{O B C D} = \int_{- 3}^{\frac{- 33}{43}} (\frac{\sqrt{3}}{2} x + \frac{3 \sqrt{3}}{2}) d x + \int_{\frac{- 33}{43}}^{0} 2 \sqrt{1 - {(\frac{x}{3})}^{2}} d x

+ \int_{0}^{\sqrt{\frac{36}{13}}} (2 \sqrt{1 - {(\frac{x}{3})}^{2}} - \sqrt{3} x) d x

Commented by TawaTawa last updated on 08/Sep/19

God bless you sir