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Question Number 49020 by ajfour last updated on 01/Dec/18

Commented by ajfour last updated on 01/Dec/18

$I f t h e i n s c r i b e d e l l i p s e i s o f m a x i m u m$ $a r e a w i t h i t s m a j o r a x i s p a r a l l e l t o$ $s i d e P Q o f △ P Q R w i t h P R = q a n d$ $Q R = p, f i n d a, b o f t h e e l l i p s e .$
Commented by ajfour last updated on 01/Dec/18

$e v e n i f a = b = R p l e a s e p r o v e s o .$
	Answered by mr W last updated on 01/Dec/18

	Commented by mr W last updated on 02/Dec/18

	$l e t m = \frac{q}{p}, λ = m^{2} - 1$ $R (0, d)$ $w h e r e d = \frac{p q}{\sqrt{p^{2} + q^{2}}} = a l t i t u d e o f t r i a n g l e$ $M (h, b) = c e n t e r o f e l l i p s e$ $e q n . o f e l l i p s e :$ $\frac{{(x - h)}^{2}}{a^{2}} + \frac{{(y - b)}^{2}}{b^{2}} = 1$ $e q n . o f R Q :$ $y = - m x + d$ $a^{2} m^{2} + b^{2} = {(d - m h - b)}^{2} \leftarrow t a n g e n t o f e l l i p s r$ $a^{2} m^{2} = {(d - m h)}^{2} - 2 (d - m h) b$ $a^{2} m^{2} + 2 (d - m h) b = {(d - m h)}^{2} . . . (i)$ $e q n . o f R P :$ $y = \frac{1}{m} x + d$ $\frac{a^{2}}{m^{2}} + b^{2} = {(d + \frac{h}{m} - b)}^{2} \leftarrow t a n g e n t o f e l l i p s e$ $\frac{a^{2}}{m^{2}} = {(d + \frac{h}{m})}^{2} - 2 (d + \frac{h}{m}) b$ $a^{2} = {(m d + h)}^{2} - 2 m (m d + h) b$ $a^{2} + 2 m (m d + h) b = {(m d + h)}^{2} . . . (i i)$ $c a s e 1 : p = q, i . e . m = 1, h = 0$ $f r o m (i) o r (i i) w e g e t$ $a^{2} + 2 d b = d^{2}$ $\Rightarrow b = \frac{d^{2} - a^{2}}{2 d}$ $a r e a o f e l l i p s e A = π a b = \frac{π (d^{2} a - a^{3})}{2 d}$ $\frac{d A}{d a} = 0 \Rightarrow d^{2} - 3 a^{2} = 0 \Rightarrow a = \frac{d}{\sqrt{3}} \Rightarrow b = \frac{d}{3}$ $c a s e 2 : p \neq q, i . e . m \neq 1$ $(i i) \times m^{2} - (i) :$ $2 [m^{4} d + m^{3} h - d + m h] b = (m^{2} d + m h + d - m h) (m^{2} d + m h - d + m h)$ $2 [(m^{2} - 1) d + m h] b = [(m^{2} - 1) d + 2 m h] d$ $\Rightarrow b = \frac{[(m^{2} - 1) d + 2 m h] d}{2 [(m^{2} - 1) d + m h]} = \frac{[(m^{2} - 1) + \frac{2 m h}{d}] d}{2 [(m^{2} - 1) + \frac{m h}{d}]}$ $\Rightarrow b = \frac{d}{2} (\frac{λ + 2 μ}{λ + μ}) w i t h μ = \frac{m h}{d} = \frac{h \sqrt{p^{2} + q^{2}}}{p^{2}}$ $(i) \times m (m d + h) - (i i) \times (d - m h) :$ $a^{2} [m^{3} (m d + h) - (d - m h)] = m (m d + h) {(d - m h)}^{2} - (d - m h) {(m d + h)}^{2}$ $a^{2} [(m^{2} - 1) d + m h] = (m h - d) (m d + h) h$ $\Rightarrow a^{2} = \frac{(m h - d) (m d + h) h}{(m^{2} - 1) d + m h} = \frac{d^{2}}{m^{2}} \times \frac{(\frac{m h}{d} - 1) (m^{2} + \frac{m h}{d}) \frac{m h}{d}}{[(m^{2} - 1) + \frac{m h}{d}]}$ $\Rightarrow a^{2} = \frac{d^{2}}{m^{2}} \times \frac{μ (μ - 1) (m^{2} + μ)}{λ + μ}$ $P = \frac{4 m^{2}}{d^{2}} \times a^{2} b^{2} = \frac{μ (μ - 1) (λ + 1 + μ) {(λ + 2 μ)}^{2}}{{(λ + μ)}^{3}}$ $s i n c e a r e a o f e l l i p s e i s A = π a b, m a x . A$ $m e a n s a l s o m a x . P .$ $\frac{d P}{d μ} = \frac{(μ - 1) (λ + 1 + μ) {(λ + 2 μ)}^{2} + μ (λ + 1 + μ) {(λ + 2 μ)}^{2} + μ (μ - 1) {(λ + 2 μ)}^{2} + 4 μ (μ - 1) (λ + 1 + μ) (λ + 2 μ)}{{(λ + μ)}^{3}} - \frac{3 μ (μ - 1) (λ + 1 + μ) {(λ + 2 μ)}^{2}}{{(λ + μ)}^{4}} = 0$ $6 μ^{4} + 13 λ μ^{3} + (9 λ^{2} - 2 λ - 2) μ^{2} + (2 λ^{3} - 3 λ^{2} - 3 λ) μ - (λ^{3} + λ^{2}) = 2 μ^{4} + λ μ^{3} - (λ^{2} + 2 λ + 2) μ^{2} + (λ^{2} + λ) μ$ $\Rightarrow 4 μ^{4} + 12 λ μ^{3} + 10 λ^{2} μ^{2} + 2 λ (λ^{2} - 2 λ - 2) μ - λ^{2} (λ + 1) = 0$ $w e s o l v e t h i s e q n . f o r μ = f (λ)$ $\Rightarrow a = \frac{d}{m} \sqrt{\frac{μ (μ - 1) (m^{2} + μ)}{m^{2} + μ - 1}}$ $\Rightarrow b = \frac{d}{2} (1 + \frac{μ}{m^{2} + μ - 1})$ $\Rightarrow h = \frac{μ d}{m} = \frac{μ p^{2}}{\sqrt{p^{2} + q^{2}}}$
	Commented by ajfour last updated on 02/Dec/18

	$h o w c a n w e f i n d h (t h e x c o o r d i n a t e$ $o f t h e c e n t e r o f e l l i p s e i n s c r i b e d$ $i n t e r m s o f a a n d b, S i r ?$
	Commented by mr W last updated on 02/Dec/18

	Commented by mr W last updated on 02/Dec/18

	$e v e n f o r t h e c a s e w i t h p = q, t h e m a x .$ $e l l i p s e i s n o t a c i r c l e .$ $m a x . A_{e l l i p s e} = \frac{π d^{2}}{3 \sqrt{3}} = \frac{π p^{2}}{6 \sqrt{3}} = 0.096 π p^{2}$ $r_{i n c i r c l e} = \frac{p}{2 + \sqrt{2}}$ $A_{i n c i r c l e} = \frac{π p^{2}}{{(2 + \sqrt{2})}^{2}} = \frac{π p^{2}}{2 (3 + 2 \sqrt{2})} = 0.085 π p^{2}$
	Commented by mr W last updated on 02/Dec/18

	Commented by mr W last updated on 02/Dec/18

	Commented by mr W last updated on 02/Dec/18

	$T o M J S S i r :$ $t h e e q n .$ $4 μ^{4} + 12 λ μ^{3} + 10 λ^{2} μ^{2} + 2 λ (λ^{2} - 2 λ - 2) μ - λ^{2} (λ + 1) = 0$ $h a s t w o o r f o u r s o l u t i o n s . c a n w e g e t t h e m$ $a c c u r a t e l y, I m e a n i n t e r m s o f λ ?$
	Commented by ajfour last updated on 02/Dec/18

	$V e r y G r e a t, S i r! y o u m a k e m e$ $v e r y h a p p y, m y {m i n d}^{'} s e y e r i g h t a w a y$ $s a i d, i t s h o u l d b e s o . (m o r e s p a c e$ $d o w n w a r d s . .)$ $T h a n k y o u s i r!$
	Commented by mr W last updated on 02/Dec/18

	$T o a j f o u r s i r :$ $μ = \frac{m h}{d} \Rightarrow h = \frac{μ d}{m} = \frac{μ p^{2}}{\sqrt{p^{2} + q^{2}}}$
	Commented by mr W last updated on 02/Dec/18

	$o r$ $f r o m b = \frac{d}{2} (\frac{λ + 2 μ}{λ + μ})$ $\Rightarrow μ = \frac{λ (2 b - d)}{2 (d - b)} = \frac{(m^{2} - 1) (2 b - d)}{2 (d - b)}$ $\Rightarrow h = \frac{(2 b - d) (q^{2} - p^{2})}{2 (d - b) \sqrt{p^{2} + q^{2}}}$
	Commented by MJS last updated on 02/Dec/18

	$sorry just read this, I^{'} ll try later$
	Commented by MJS last updated on 02/Dec/18

	$I^{'} m afraid {it}^{'} s not possible to solve in terms$ $of λ . . .$
	Commented by mr W last updated on 03/Dec/18

	$t h a n k y o u a n y w a y s i r!$ ${i t}^{'} s t o o c o m p l i c a t e d i n d e e d .$
	Answered by ajfour last updated on 03/Dec/18

	$I f R (0, H) a n d p \neq q$ $T h e n a a n d b o f e l l i p s e a r e r e l a t e d$ $b y t h e f o l l o w i n g e x p r e s s i o n :$ $(H - b) (p^{2} + q^{2})$ $= p \sqrt{b^{2} p^{2} + a^{2} q^{2}} + q \sqrt{b^{2} q^{2} + a^{2} p^{2}} .$ $I h o w e v e r f i n d i t d i f f i c u l t t o$ $m a x i m i s e a^{2} b^{2} .$ $A n d i f p = q, t h e n$ $b = \frac{H}{3}, a = \frac{H}{\sqrt{3}} w h e r e H = \frac{p q}{\sqrt{p^{2} + q^{2}}} .$
	Commented by mr W last updated on 03/Dec/18
	$(H−b)(p^2 +q^2 )= p (√(b^2 p^2 +a^2 q^2 ))+q (√(b^2 q^2 +a^2 p^2 )) (aH−ab)(p^2 +q^2 )= p (√(a^2 b^2 p^2 +a^4 q^2 ))+q (√(a^2 b^2 q^2 +a^4 p^2 )) (aH−S)(p^2 +q^2 )= p (√(S^2 p^2 +a^4 q^2 ))+q (√(S^2 q^2 +a^4 p^2 )) with S=ab (dS/da)=0 H(p^2 +q^2 )= p ((4a^3 q^2 )/(2(√(S^2 p^2 +a^4 q^2 ))))+q((4a^3 p^2 )/(2 (√(S^2 q^2 +a^4 p^2 )) )) H(p^2 +q^2 )=2pqa^2 [ (q/(√(b^2 p^2 +a^2 q^2 )))+(p/( (√(b^2 q^2 +a^2 p^2 )) ))] H(p^2 +q^2 )=2pqa^2 [ ((q (√(b^2 q^2 +a^2 p^2 ))+p(√(b^2 p^2 +a^2 q^2 )))/(√((b^2 p^2 +a^2 q^2 )(b^2 q^2 +a^2 p^2 ))))] H(p^2 +q^2 )=2pqa^2 [ (((H−b)(p^2 +q^2 ))/(√((b^2 p^2 +a^2 q^2 )(b^2 q^2 +a^2 p^2 ))))] H=2pqa^2 [ (((H−b))/(√((b^2 p^2 +a^2 q^2 )(b^2 q^2 +a^2 p^2 ))))] let β=(b/a) ((pq)/(√(p^2 +q^2 )))=2pq[ (((H−b))/(√((β^2 p^2 +q^2 )(β^2 q^2 +p^2 ))))] (1/(√(p^2 +q^2 )))=2[ (((H−b))/(√((β^2 p^2 +q^2 )(β^2 q^2 +p^2 ))))] b=H[1−(1/(2H))(√(((β^2 p^2 +q^2 )(β^2 q^2 +p^2 ))/(p^2 +q^2 )))] (H/b)=(1/([1−(1/(2H))(√(((β^2 p^2 +q^2 )(β^2 q^2 +p^2 ))/(p^2 +q^2 )))])) =(1/(1−((√((β^2 p^2 +q^2 )(β^2 q^2 +p^2 )))/(2pq)))) ((H/b)−1)(p^2 +q^2 )β= p (√(β^2 p^2 +q^2 ))+q (√(β^2 q^2 +p^2 )) {(1/(1−((√((β^2 p^2 +q^2 )(β^2 q^2 +p^2 )))/(2pq))))−1}(p^2 +q^2 )β= p (√(β^2 p^2 +q^2 ))+q (√(β^2 q^2 +p^2 )) with m=(q/p) ⇒{(1/(1−((√((β^2 +m^2 )(β^2 m^2 +1)))/2)))−1}(1+m^2 )β= (√(β^2 +m^2 ))+m (√(β^2 m^2 +1)) ⇒β=....$
	$(H - b) (p^{2} + q^{2}) = p \sqrt{b^{2} p^{2} + a^{2} q^{2}} + q \sqrt{b^{2} q^{2} + a^{2} p^{2}}$ $(a H - a b) (p^{2} + q^{2}) = p \sqrt{a^{2} b^{2} p^{2} + a^{4} q^{2}} + q \sqrt{a^{2} b^{2} q^{2} + a^{4} p^{2}}$ $(a H - S) (p^{2} + q^{2}) = p \sqrt{S^{2} p^{2} + a^{4} q^{2}} + q \sqrt{S^{2} q^{2} + a^{4} p^{2}}$ $w i t h S = a b$ $\frac{d S}{d a} = 0$ $H (p^{2} + q^{2}) = p \frac{4 a^{3} q^{2}}{2 \sqrt{S^{2} p^{2} + a^{4} q^{2}}} + q \frac{4 a^{3} p^{2}}{2 \sqrt{S^{2} q^{2} + a^{4} p^{2}}}$ $H (p^{2} + q^{2}) = 2 {p q a}^{2} [\frac{q}{\sqrt{b^{2} p^{2} + a^{2} q^{2}}} + \frac{p}{\sqrt{b^{2} q^{2} + a^{2} p^{2}}}]$ $H (p^{2} + q^{2}) = 2 {p q a}^{2} [\frac{q \sqrt{b^{2} q^{2} + a^{2} p^{2}} + p \sqrt{b^{2} p^{2} + a^{2} q^{2}}}{\sqrt{(b^{2} p^{2} + a^{2} q^{2}) (b^{2} q^{2} + a^{2} p^{2})}}]$ $H (p^{2} + q^{2}) = 2 {p q a}^{2} [\frac{(H - b) (p^{2} + q^{2})}{\sqrt{(b^{2} p^{2} + a^{2} q^{2}) (b^{2} q^{2} + a^{2} p^{2})}}]$ $H = 2 {p q a}^{2} [\frac{(H - b)}{\sqrt{(b^{2} p^{2} + a^{2} q^{2}) (b^{2} q^{2} + a^{2} p^{2})}}]$ $l e t β = \frac{b}{a}$ $\frac{p q}{\sqrt{p^{2} + q^{2}}} = 2 p q [\frac{(H - b)}{\sqrt{(β^{2} p^{2} + q^{2}) (β^{2} q^{2} + p^{2})}}]$ $\frac{1}{\sqrt{p^{2} + q^{2}}} = 2 [\frac{(H - b)}{\sqrt{(β^{2} p^{2} + q^{2}) (β^{2} q^{2} + p^{2})}}]$ $b = H [1 - \frac{1}{2 H} \sqrt{\frac{(β^{2} p^{2} + q^{2}) (β^{2} q^{2} + p^{2})}{p^{2} + q^{2}}}]$ $\frac{H}{b} = \frac{1}{[1 - \frac{1}{2 H} \sqrt{\frac{(β^{2} p^{2} + q^{2}) (β^{2} q^{2} + p^{2})}{p^{2} + q^{2}}}]}$ $= \frac{1}{1 - \frac{\sqrt{(β^{2} p^{2} + q^{2}) (β^{2} q^{2} + p^{2})}}{2 p q}}$ $(\frac{H}{b} - 1) (p^{2} + q^{2}) β = p \sqrt{β^{2} p^{2} + q^{2}} + q \sqrt{β^{2} q^{2} + p^{2}}$ ${\frac{1}{1 - \frac{\sqrt{(β^{2} p^{2} + q^{2}) (β^{2} q^{2} + p^{2})}}{2 p q}} - 1} (p^{2} + q^{2}) β = p \sqrt{β^{2} p^{2} + q^{2}} + q \sqrt{β^{2} q^{2} + p^{2}}$ $w i t h m = \frac{q}{p}$ $\Rightarrow {\frac{1}{1 - \frac{\sqrt{(β^{2} + m^{2}) (β^{2} m^{2} + 1)}}{2}} - 1} (1 + m^{2}) β = \sqrt{β^{2} + m^{2}} + m \sqrt{β^{2} m^{2} + 1}$ $\Rightarrow β = . . . .$
	Commented by mr W last updated on 03/Dec/18

	$s i r, I c h a n g e d y o u r ‘ ‘ -^{″} s i g n t o ‘ ‘ +^{″} w i t h o u t$ $t o k n o w i f i t i s c o r r e c t . d u e t o s y m m e t r y$ $I t h i n k i t s h o u l d b e ‘ ‘ +^{″}, b e c a u s e t h e$ $r e s u l t s h o u l d b e u n c h a n g e d i f w e$ $e x c h a n g e p a n d q .$
	Commented by ajfour last updated on 03/Dec/18

	$y e s s i r, i t s h o u l d b e^{'} +^{'} .$ $t h a n k s f o r p o i n t i n g o u t, a n d a l l$ $t h e w o r k i n g s t h e r e a f t e r . .$
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