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Question Number 39379 by rahul 19 last updated on 05/Jul/18

Commented by MJS last updated on 06/Jul/18
$(1) ∣∣x∣−∣y∣∣=1 (2) 2a∣x∣∣y∣+1=2∣x∣+a∣y∣ (1) { ((y_1 =−∣x∣−1; range(y_1 )=]−∞; −1])),((y_2 =∣x∣+1; range(y_2 )=[1; +∞[)),((∣x∣≥1 { ((y_3 =−∣x∣+1; range(y_3 )=]−∞; 0])),((y_4 =∣x∣−1; range(y_4 )=[0; +∞[)) :})) :} (2) 2a∣x∣∣y∣−a∣y∣=2∣x∣−1 a∣y∣(2∣x∣−1)=2∣x∣−1 ∣y∣=(1/a) ⇒ { ((y_5 =−(1/a))),((y_6 =(1/a))) :} because range of y_1 and y_2 excludes ]−1; 1[ and they reach −1 and 1 respectively only once ⇒ ∣(1/a)∣>1 ⇒ ∣a∣<1$
$(1) ∣∣ x ∣ - ∣ y ∣∣= 1$ $(2) 2 a ∣ x ∣∣ y ∣ + 1 = 2 ∣ x ∣ + a ∣ y ∣$ $(1) {\begin{cases} y_{1} = - ∣ x ∣ - 1; range (y_{1}) =] - \infty; - 1] \\ y_{2} =∣ x ∣ + 1; range (y_{2}) = [1; + \infty [ \\ ∣ x ∣⩾ 1 {\begin{cases} y_{3} = - ∣ x ∣ + 1; range (y_{3}) =] - \infty; 0] \\ y_{4} =∣ x ∣ - 1; range (y_{4}) = [0; + \infty [ \end{cases} \end{cases}$ $(2) 2 a ∣ x ∣∣ y ∣ - a ∣ y ∣= 2 ∣ x ∣ - 1$ $a ∣ y ∣ (2 ∣ x ∣ - 1) = 2 ∣ x ∣ - 1$ $∣ y ∣= \frac{1}{a} \Rightarrow {\begin{cases} y_{5} = - \frac{1}{a} \\ y_{6} = \frac{1}{a} \end{cases}$ $because range of y_{1} and y_{2} excludes] - 1; 1 [$ $and they reach - 1 and 1 respectively only$ $once \Rightarrow ∣ \frac{1}{a} ∣> 1 \Rightarrow ∣ a ∣< 1$
Commented by MJS last updated on 06/Jul/18

$a is not a number but its range is] - 1; 1 [, so$ $- 3 < 3 a < 3 we {can}^{'} t give a single value$
Commented by rahul 19 last updated on 06/Jul/18
What's the value of 3a ?
Commented by rahul 19 last updated on 06/Jul/18
Ans. given is 2!
Commented by MJS last updated on 06/Jul/18

$draw these functions, I think the given answer$ $is wrong . or else we must limit the allowed$ $range of a (but {there}^{'} s no such limit given)$
Commented by rahul 19 last updated on 06/Jul/18
Ok sir ! thanks ! ��
	Answered by tanmay.chaudhury50@gmail.com last updated on 06/Jul/18

	$2 a ∣ x ∣ ∣ y ∣ + 1 - 2 ∣ x ∣ - a ∣ y ∣ = 0$ $2 ∣ x ∣ {a ∣ y ∣ - 1} - {a ∣ y ∣ - 1} = 0$ ${a ∣ y ∣ - 1} {2 ∣ x ∣ - 1} = 0$ $∣ x ∣= \frac{1}{2} ∣ y ∣= \frac{1}{a}$ $a t t a c h i n g t b e g r a p h o f$ $1 . ∣∣ x ∣ - ∣ y ∣∣= 1$ $2 . ∣ x ∣ = \frac{1}{2}$ $3 . ∣ y ∣ = \frac{1}{a} a < 1$ $p l s w a i t . . .$
	Commented by tanmay.chaudhury50@gmail.com last updated on 06/Jul/18

	Commented by tanmay.chaudhury50@gmail.com last updated on 06/Jul/18

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