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Question Number 27818 by ajfour last updated on 15/Jan/18

Commented by ajfour last updated on 15/Jan/18

$I f y = {A x}^{2} i s r e f l e c t e d i n t h e l i n e$ $\frac{x}{a} + \frac{y}{b} = 1, f i n d e q u a t i o n o f i t s$ $r e f l e c t i o n f (x, y) = 0 .$
Commented by mrW2 last updated on 15/Jan/18

${l e t}^{'} s s a y p o i n t A (p, q) i s r e f l e c t e d i n$ $l i n e \frac{x}{a} + \frac{y}{b} = 1 a n d i t s m i r r o r e d i m a g e$ $i s p o i n t C (u, v) .$ ${l e t}^{'} s s a y B i s a p o i n t o n t h e l i n e w i t h$ $A B ⊥ t o t h e l i n e .$ $E q n . o f A B i s$ $(x, y) = (p, r) + λ (\frac{1}{a}, \frac{1}{b})$ $o r$ $x = p + \frac{λ}{a}$ $y = q + \frac{λ}{b}$ $p o i n t B i s o n t h e l i n e :$ $\frac{1}{a} (p + \frac{λ_{B}}{a}) + \frac{1}{b} (q + \frac{λ_{B}}{b}) = 1$ $λ_{B} (\frac{1}{a^{2}} + \frac{1}{b^{2}}) = 1 - (\frac{p}{a} + \frac{q}{b})$ $λ_{B} (\frac{a^{2} + b^{2}}{a b}) = a b - (a q + b p)$ $\Rightarrow λ_{B} = \frac{a b (a b - a q - b p)}{a^{2} + b^{2}}$ $s i n c e C i s m i r r o r e d i m a g e o f A,$ $λ_{C} = 2 λ_{B} = \frac{2 a b (a b - a q - b p)}{a^{2} + b^{2}}$ $\Rightarrow u = p + \frac{λ_{C}}{a} = p + \frac{2 b (a b - a q - b p)}{a^{2} + b^{2}} = \frac{a^{2} - b^{2}}{a^{2} + b^{2}} p - \frac{2 a b}{a^{2} + b^{2}} q + \frac{2 {a b}^{2}}{a^{2} + b^{2}} = c p + d q - d b$ $\Rightarrow v = q + \frac{λ_{C}}{b} = q + \frac{2 a (a b - a q - b p)}{a^{2} + b^{2}} = - \frac{2 a b}{a^{2} + b^{2}} p + \frac{b^{2} - a^{2}}{a^{2} + b^{2}} q + \frac{2 a^{2} b}{a^{2} + b^{2}} = d p - c q - d a$ $w i t h c = \frac{a^{2} - b^{2}}{a^{2} + b^{2}} a n d d = - \frac{2 a b}{a^{2} + b^{2}}$ $n o w w e h a v e (u, v) o n t h e c u r v e y = {A x}^{2}$ $\Rightarrow v = {A u}^{2}$ $d p - c q - d a = A {(c p + d q - d b)}^{2} = A (c^{2} p^{2} + 2 c d p q + d^{2} q^{2} + d^{2} b^{2} - 2 d b c p - 2 d^{2} b q)$ ${A c}^{2} p^{2} + {A d}^{2} q^{2} + 2 A c d p q + {A d}^{2} b^{2} - 2 A d b c p - 2 {A d}^{2} b q - d p + c q + d a = 0$ ${A c}^{2} p^{2} + {A d}^{2} q^{2} + 2 A c d p q - d (1 + 2 A b c) p - (2 {A b d}^{2} - c) q + d (a + {A d b}^{2}) = 0$ $o r$ ${A c}^{2} x^{2} + {A d}^{2} y^{2} + 2 A c d x y - d (1 + 2 A b c) x - (2 {A b d}^{2} - c) y + d (a + {A d b}^{2}) = 0$ $\frac{A {(a^{2} - b^{2})}^{2}}{a^{2} + b^{2}} x^{2} + \frac{4 {A a}^{2} b^{2}}{a^{2} + b^{2}} y^{2} - \frac{4 A a b (a^{2} - b^{2})}{a^{2} + b^{2}} x y + 2 a b (1 + \frac{2 A b (a^{2} - b^{2})}{a^{2} + b^{2}}) x - (\frac{8 {A a}^{2} b^{3}}{a^{2} + b^{2}} - a^{2} + b^{2}) y - 2 a b (a - \frac{2 {A a b}^{3}}{a^{2} + b^{2}}) = 0$ $\Rightarrow f (x, y) = A {(a^{2} - b^{2})}^{2} x^{2} + 4 {A a}^{2} b^{2} y^{2} - 4 A a b (a^{2} - b^{2}) x y + 2 a b (a^{2} + b^{2} + 2 {A a}^{2} b - 2 {A b}^{3}) x - (8 {A a}^{2} b^{3} - a^{4} + b^{4}) y - 2 a b (a^{3} + {a b}^{2} - 2 {A a b}^{3}) = 0$
Commented by mrW2 last updated on 15/Jan/18

Commented by ajfour last updated on 15/Jan/18

$G r e a t S i r! t h i s i s t r u l y a m a z i n g .$ $i s h a l l t r y t o f o l l o w, m i g h t t a k e a w h i l e .$
Commented by mrW2 last updated on 15/Jan/18

Commented by mrW2 last updated on 15/Jan/18

$T h i s i s a l s o a g o o d w a y i n c u r r e n t c a s e,$ $b e c a u s e t h e c u r v e t o r e f l e c t i s a$ $s y m m e t r i c f i g u r e . I f t h e f i g u r e i s$ $n o t s y m m e t r i c, e . g . i f y = e^{x}, y o u c a n$ $n o t o b t a i n t h e m i r r o r e d f u n c t i o n$ $t h r o u g h s u c h a r o t a t i o n .$
Commented by mrW2 last updated on 16/Jan/18

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