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Question Number 46919 by behi83417@gmail.com last updated on 02/Nov/18

Commented by ajfour last updated on 02/Nov/18

Commented by ajfour last updated on 02/Nov/18

$l e t C F = h; F M = F B = \sqrt{4 - h^{2}}$ $A B = \sqrt{9 - h^{2}} + \sqrt{4 - h^{2}} = 2 M B$ $\Rightarrow \sqrt{9 - h^{2}} + \sqrt{4 - h^{2}} = 4 \sqrt{4 - h^{2}}$ $\Rightarrow 9 - h^{2} = 9 (4 - h^{2})$ $o r 8 h^{2} = 27 \Rightarrow h = \frac{3 \sqrt{3}}{2 \sqrt{2}}$ $A B = 4 \sqrt{4 - \frac{27}{8}} = \sqrt{10}$ $A r e a (△ A B C) = \frac{h}{2} \times A B$ $= \frac{3 \sqrt{3}}{4 \sqrt{2}} \times \sqrt{10} = \frac{3 \sqrt{15}}{4} .$
Commented by behi83417@gmail.com last updated on 02/Nov/18

$t h a n k y o u s o m u c h s i r A j f o u r .$ $y o u r a n s w e r i s r i g h t w i t h 13 %$ $d e f f e r e n c e .$ $a r e a = 2.76 u n i t s .$ $You can't use 'macro parameter character #' in math mode$
Commented by ajfour last updated on 02/Nov/18

$B e h i S i r, i c h e c k e d, b u t {c o u l d n}^{'} t$ $f i n d m i s t a k e i n m y s o l u t i o n . .!$
Commented by MJS last updated on 02/Nov/18

$A M = 2 m$ $M F = F B = m$ $m^{2} + h^{2} = 4$ $9 m^{2} + h^{2} = 9$ $\Rightarrow m = \frac{\sqrt{10}}{4} \land h = \frac{3 \sqrt{6}}{4}$ $\Rightarrow c = A B = 4 m = \sqrt{10}$ $\Rightarrow area = \frac{3 \sqrt{15}}{4}$ $. . . so Sir Aifour is right$
Commented by behi83417@gmail.com last updated on 02/Nov/18

$t h a n k y o u v e r y m u c h s i r .$ $y o u a n d s i r A j f o u r, u s e :$ $[a = 2, m = 2] t o t a k e C \overset{△}{M B}, a s i s o s c a l e$ $t r i a n g l e . n o w i f a \neq 2, w h a t c a n$ $w e d o ? [s a y a = 5, f o r e x a m p l e]$ $(f o r : a = 5, b = 3, m = 2,$ $a r e a = \frac{8 \sqrt{161}}{15} = 6.77)$
Commented by behi83417@gmail.com last updated on 03/Nov/18

$2 (a^{2} + b^{2}) = c^{2} + 4 m_{c}^{2} \Rightarrow 2 (2^{2} + 3^{2}) = c^{2} + 4 \times 2^{2}$ $\Rightarrow c^{2} = 26 - 16 = 10 \Rightarrow c = \sqrt{10}$ $S = \frac{1}{4} \sqrt{(2 + 3 + \sqrt{10}) (2 + 3 - \sqrt{10}) (2 + \sqrt{10} - 3) (3 + \sqrt{10} - 2)} =$ $= \frac{1}{4} \sqrt{(5 + \sqrt{10}) (5 - \sqrt{10}) (\sqrt{10} + 1) (\sqrt{10} - 1)} =$ $= \frac{1}{4} \sqrt{(25 - 10) (10 - 1)} = \frac{3 \sqrt{15}}{4} .$ $t h e f o r m u l a g i v e n f o r i s :$ $S = \frac{m_{c} . (a + b)}{4 ab} \sqrt{4 a^{2} b^{2} - m_{c}^{2} . {(a + b)}^{2}}$ $but i {can}^{'} t p r o v e i t .$
Commented by ajfour last updated on 03/Nov/18

$l e t M (0, 0); A (- l, 0); B (l, 0);$ $C (x, y)$ $\Rightarrow S = \frac{1}{2} (2 l) y = l y$ $\Rightarrow x^{2} + y^{2} = m^{2}$ ${(x + l)}^{2} + y^{2} = b^{2}$ ${(x - l)}^{2} + y^{2} = a^{2}$ $\Rightarrow 2 l^{2} = a^{2} + b^{2} - 2 m^{2}$ $& 4 l x = b^{2} - a^{2}$ $\Rightarrow 16 l^{2} x^{2} = {(b^{2} - a^{2})}^{2}$ $\Rightarrow 16 l^{2} (m^{2} - y^{2}) = {(b^{2} - a^{2})}^{2}$ $o r 16 l^{2} m^{2} - 16 S^{2} = {(b^{2} - a^{2})}^{2}$ $S^{2} = \frac{16 l^{2} m^{2} - {(b^{2} - a^{2})}^{2}}{16}$ $= \frac{8 m^{2} (a^{2} + b^{2} - 2 m^{2}) - {(b^{2} - a^{2})}^{2}}{16}$ $= \frac{8 m^{2} (a^{2} + b^{2}) - {(a^{2} + b^{2})}^{2} + 4 a^{2} b^{2} - 16 m^{4}}{16}$ $= \frac{4 a^{2} b^{2} - {(a^{2} + b^{2} - 4 m^{2})}^{2}}{16}$ $\Rightarrow f o r a = 2, b = 3, m = 2$ $S^{2} = \frac{144 - 9}{16} = \frac{135}{16}$ $S = \frac{3 \sqrt{15}}{4} .$ $I f b = 3, a = 5, m = 2, t h e n$ $S^{2} = \frac{900 - 324}{16} = \frac{576}{16} = 36$ $S = 6 .$
Commented by behi83417@gmail.com last updated on 03/Nov/18

$s i r A j f o u r . y o u a n d y o u r s o l u t i o n$ $a r e a m a z i n g . t h a n k y o u s o m u c h .$
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