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Question Number 60010 by naka3546 last updated on 17/May/19

Commented by naka3546 last updated on 17/May/19

$F O E c o l l i n e a r ?$
	Answered by tanmay last updated on 17/May/19
	$CD×CE=CH^2 (CE+ED)×CE=243 circle(x−5)^2 +(y−5)^2 =5^2 BC=5+9(√3) tan30^o =((AB)/(BC))→AB=(5+9(√3) )×(1/(√3)) eqn AC (x/(5+9(√3)))+((y(√3) )/((5+9(√3) )))=1 x+y(√3) −(5+9(√3) )=0 distance from (5,5) to st line AC is d(say) d=∣((5+5(√3) −5−9(√3))/(√(1^2 +((√3) )^2 )))∣=2(√3) (2(√3) )^2 +(((DE)/2))^2 =5^2 DE^2 =52 DE=2(√(13)) (CE+DE)×CE=243 (CE+2(√(13)) )×CE=243 CE^2 +2×CE×(√(13)) +13=256 (CE+(√(13)) )^2 =16^2 CE=16+(√(13)) ,16−(√(13)) let coordinate of E(a,b) sin30^o =(b/(16+(√(13)))) [or(b/(16−(√(13))))] b=(1/2)(16+(√(13)) ) or b=(1/2)(16−(√(13)) ) cos30^o =((√3)/2)=(a/(16+(√(13)))) a=((√3)/2)(16+(√(13 )) ) eqn of st line FO is y=5 but {((√3)/2)(16+(√(13)) ),((16+(√(13)))/2)) do not satisy eqn y=5 so F,O,E not colliniar... pls check mistake if any$
	$C D \times C E = {C H}^{2}$ $(C E + E D) \times C E = 243$ $c i r c l e {(x - 5)}^{2} + {(y - 5)}^{2} = 5^{2}$ $B C = 5 + 9 \sqrt{3}$ $t a n 30^{o} = \frac{A B}{B C} \to A B = (5 + 9 \sqrt{3}) \times \frac{1}{\sqrt{3}}$ $e q n A C \frac{x}{5 + 9 \sqrt{3}} + \frac{y \sqrt{3}}{(5 + 9 \sqrt{3})} = 1$ $x + y \sqrt{3} - (5 + 9 \sqrt{3}) = 0$ $d i s t a n c e f r o m (5, 5) t o s t l i n e A C i s d (s a y)$ $d =∣ \frac{5 + 5 \sqrt{3} - 5 - 9 \sqrt{3}}{\sqrt{1^{2} + {(\sqrt{3})}^{2}}} ∣= 2 \sqrt{3}$ ${(2 \sqrt{3})}^{2} + {(\frac{D E}{2})}^{2} = 5^{2}$ ${D E}^{2} = 52$ $D E = 2 \sqrt{13}$ $(C E + D E) \times C E = 243$ $(C E + 2 \sqrt{13}) \times C E = 243$ ${C E}^{2} + 2 \times C E \times \sqrt{13} + 13 = 256$ ${(C E + \sqrt{13})}^{2} = 16^{2}$ $C E = 16 + \sqrt{13}, 16 - \sqrt{13}$ $l e t c o o r d i n a t e o f E (a, b)$ $s i n 30^{o} = \frac{b}{16 + \sqrt{13}} [o r \frac{b}{16 - \sqrt{13}}]$ $b = \frac{1}{2} (16 + \sqrt{13}) o r b = \frac{1}{2} (16 - \sqrt{13})$ $c o s 30^{o} = \frac{\sqrt{3}}{2} = \frac{a}{16 + \sqrt{13}}$ $a = \frac{\sqrt{3}}{2} (16 + \sqrt{13})$ $e q n o f s t l i n e F O i s y = 5$ $b u t {\frac{\sqrt{3}}{2} (16 + \sqrt{13}), \frac{16 + \sqrt{13}}{2}) d o n o t s a t i s y e q n$ $y = 5$ $s o F, O, E n o t c o l l i n i a r . . .$ $p l s c h e c k m i s t a k e i f a n y$
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