A-linear-transformation-E-of-the-x-y-plane-is-defined-as-E-x-y-2x-y-2x-3y-Find-the-equation-of-the-line-that-remains-invariant-under-the-transformation-

Question Number 183965 by nadovic last updated on 01/Jan/23

A linear transformation E, of the

x - y plane is defined as

E : (x, y) \to (2 x + y, 2 x + 3 y)

Find the equation of the line that

remains invariant under the

transformation .

Answered by mr W last updated on 01/Jan/23

s a y t h e l i n e i s :

a x + b y + c = 0

u n d e r t r a n s f o r m a t i o n :

a (2 x + y) + b (2 x + 3 y) + c = 0

\Rightarrow (2 a + 2 b) x + (a + 3 b) y + c = 0

\underset{―}{i f c \neq 0 :}

\frac{2 a + 2 b}{c} = \frac{a}{c}

b \frac{a + 3 b}{c} = \frac{b}{c}

\Rightarrow a = - 2 b

\Rightarrow - 2 b x + b y + c = 0

o r 2 x - y + k = 0 ✓

\underset{―}{i f c = 0 :}

\frac{2 a + 2 b}{a} = \frac{a + 3 b}{b}

2 {(\frac{b}{a})}^{2} - (\frac{b}{a}) - 1 = 0

(2 \frac{b}{a} + 1) (\frac{b}{a} - 1) = 0

\Rightarrow \frac{b}{a} = - \frac{1}{2} o r \frac{b}{a} = 1

\Rightarrow 2 x - y = 0 (i n c l u d e d i n 2 x - y + k = 0)

o r x + y = 0 ✓

\underset{―}{s u m m a r y :}

t h e s e a r c h e d l i n e i s

2 x - y + k = 0 o r

x + y = 0

Commented by nadovic last updated on 01/Jan/23

T h a n k s S i r, b u t t h e b o o k g a v e t w o

a n s w e r s : 2 x - y + c = 0 a n d y = - x + c

Commented by mr W last updated on 01/Jan/23

y = - x + c i s n o t c o r r e c t . b u t y = - x i s

o k . s e e a b o v e .

Commented by nadovic last updated on 01/Jan/23

Alright . . Thank you very much .

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