Question-65281

Question Number 65281 by mr W last updated on 27/Jul/19

Commented by mr W last updated on 27/Jul/19

s e g m e n t o f c i r c l e w i t h s i z e 2 c \times d . (d ⩽ c)

f i n d t h e i n s c r i b e d e l l i p s e w i t h

m a x i m u m a r e a .

Answered by mr W last updated on 28/Jul/19

R = r a d i u s o f c i r c l e

d (2 R - d) = c^{2}

\Rightarrow R = \frac{c^{2} + d^{2}}{2 d}

l e t h = R - d = \frac{c^{2} - d^{2}}{2 d}

e q n . o f c i r c l e :

x^{2} + {(y + h)}^{2} = R^{2}

x^{2} + y^{2} + 2 h y + h^{2} - R^{2} = 0

\Rightarrow x^{2} + y^{2} + 2 h y - c^{2} = 0 \dots (i)

e q n . o f e l l i p s e :

\frac{x^{2}}{a^{2}} + \frac{{(y - b)}^{2}}{b^{2}} = 1

x^{2} + \frac{a^{2}}{b^{2}} y^{2} - \frac{2 a^{2}}{b} y = 0 \dots (i i)

(i) - (i i) :

(1 - \frac{a^{2}}{b^{2}}) y^{2} + 2 (h + \frac{a^{2}}{b}) y - c^{2} = 0

d u e t o t a n g e n c y t h e r e i s o n l y o n e r o o t,

Δ = 4 {(h + \frac{a^{2}}{b})}^{2} + 4 c^{2} (1 - \frac{a^{2}}{b^{2}}) = 0

\Rightarrow {(h b + a^{2})}^{2} + c^{2} (b^{2} - a^{2}) = 0 \dots (i i i)

\Rightarrow {(h a b + a^{3})}^{2} + c^{2} (a^{2} b^{2} - a^{4}) = 0

l e t P = a b

\Rightarrow {(h P + a^{3})}^{2} + c^{2} (P^{2} - a^{4}) = 0

f o r m a x i m u m e l l i p s e a r e a, \frac{d P}{d a} = 0

\Rightarrow 2 (h P + a^{3}) (3 a^{2}) + c^{2} (- 4 a^{3}) = 0

\Rightarrow 3 (h b + a^{2}) - 2 c^{2} = 0

i f h = 0, i . e . d = c :

3 a^{2} - 2 c^{2} = 0 \Rightarrow a = \frac{\sqrt{6} c}{3} \Rightarrow b = \frac{\sqrt{2} c}{3}

i f h \neq 0 :

\Rightarrow b = \frac{1}{h} (\frac{2 c^{2}}{3} - a^{2})

i n s e r t t h i s i n t o (i i i) :

{(\frac{2 c^{2}}{3} - a^{2} + a^{2})}^{2} + c^{2} (b^{2} - a^{2}) = 0

\frac{4 c^{2}}{9} + \frac{1}{h^{2}} {(\frac{2 c^{2}}{3} - a^{2})}^{2} - a^{2} = 0

\frac{4 c^{2} h^{2}}{9} + {(\frac{2 c^{2}}{3} - a^{2})}^{2} - h^{2} a^{2} = 0

a^{4} - (\frac{4 c^{2}}{3} + h^{2}) a^{2} + \frac{4 c^{2} (c^{2} + h^{2})}{9} = 0

\Rightarrow a^{2} = \frac{1}{2} [\frac{4 c^{2}}{3} + h^{2} - \sqrt{{(\frac{4 c^{2}}{3} + h^{2})}^{2} - \frac{16 c^{2} (c^{2} + h^{2})}{9}}]

\Rightarrow a^{2} = \frac{1}{2} [\frac{4 c^{2}}{3} + h^{2} - \sqrt{\frac{16 c^{4} + 24 c^{2} h^{2} + 9 h^{4} - 16 c^{4} - 16 c^{2} h^{2}}{9}}]

\Rightarrow a^{2} = \frac{1}{2} (\frac{4 c^{2}}{3} + h^{2} - h \sqrt{\frac{8 c^{2}}{9} + h^{2}})

\Rightarrow a = \sqrt{\frac{1}{2} (\frac{4 c^{2}}{3} + h^{2} - h \sqrt{\frac{8 c^{2}}{9} + h^{2}})}

Commented by mr W last updated on 28/Jul/19

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