find-for-equation-of-image-ellipse-x-2-9-y-2-8-1-if-reflected-with-line-x-y-4-

Question Number 84531 by jagoll last updated on 14/Mar/20

find for equation of image ellipse

\frac{x^{2}}{9} + \frac{y^{2}}{8} = 1 if reflected with line

x + y = - 4

Commented by jagoll last updated on 14/Mar/20

help me

Commented by mr W last updated on 14/Mar/20

i m i s r e a d t h e q u e s t i o n a s x + y = 4, b u t

t h e m e t h o d i s t h e s a m e .

Answered by mr W last updated on 14/Mar/20

i m a g e o f p o i n t (u, v) i n t h e l i n e

a x + b y + c = 0 i s p o i n t (x, y) w i t h

\frac{x - u}{a} = \frac{y - v}{b} = - \frac{2 (a u + b v + c)}{a^{2} + b^{2}}

o r

x = u - \frac{2 a (a u + b v + c)}{a^{2} + b^{2}}

y = v - \frac{2 b (a u + b v + c)}{a^{2} + b^{2}}

i n o u r c a s e : t h e l i n e i s x + y = 4,

i . e . a = b = 1, c = - 4

\Rightarrow x = 4 - v

\Rightarrow y = 4 - u

f r o m \frac{x^{2}}{9} + \frac{y^{2}}{8} = 1 w e g e t i t s i m a g e

\Rightarrow \frac{{(4 - v)}^{2}}{9} + \frac{{(4 - u)}^{2}}{8} = 1

i . e . t h e e q n . o f t h e i m a g e e l l i p s e i s

\frac{{(4 - y)}^{2}}{9} + \frac{{(4 - x)}^{2}}{8} = 1

Commented by mr W last updated on 14/Mar/20

Commented by mr W last updated on 14/Mar/20

f u r t h e r e x a m p l e s :

i m a g e o f \frac{x^{2}}{25} + \frac{y^{2}}{9} = 1 i s

\frac{{(4 - y)}^{2}}{25} + \frac{{(4 - x)}^{2}}{9} = 1

i m a g e o f y = x^{2} - 4 i s

4 - x = {(4 - y)}^{2} - 4 o r x = 8 - {(4 - y)}^{2}

i m a g e o f y = s i n x i s

4 - x = \sin (4 - y) o r x = 4 - \sin (4 - y)

Commented by mr W last updated on 14/Mar/20

Commented by mr W last updated on 14/Mar/20

Commented by mr W last updated on 14/Mar/20

Commented by jagoll last updated on 14/Mar/20

yes sir . i got result \frac{{(x - 4)}^{2}}{9} + \frac{{(y - 4)}^{2}}{8} = 1

i will learn your method sir .

thank you very much

Commented by jagoll last updated on 14/Mar/20

sir x = 4 - v, where your got 4 sir

Commented by mr W last updated on 14/Mar/20

e q n . o f r e f l e c t i o n l i n e : x + y - 4 = 0

w i t h a = b = 1, c = - 4

x = u - \frac{2 a (a u + b v + c)}{a^{2} + b^{2}} = u - \frac{2 (u + v - 4)}{2} = 4 - v

y = v - \frac{2 b (a u + b v + c)}{a^{2} + b^{2}} = v - \frac{2 (u + v - 4)}{2} = 4 - u

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