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 and Answers Forum

All Questions Topic List
Matrices and Determinants Questions
Previous in All Question Next in All Question
Previous in Matrices and Determinants Next in Matrices and Determinants
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

	$Alright . . Thank you very much .$
Terms of Service
Privacy Policy
Contact: info@tinkutara.com

Question and Answers Forum