If , x^( 2) + 9y^( 2) + 4x +18y −23=0 then find the value of , M


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Question Number 161914 by mnjuly1970 last updated on 24/Dec/21

$If,$ $x^{2} + 9 y^{2} + 4 x + 18 y - 23 = 0$ $t h e n f i n d t h e v a l u e o f, M_{} a x (3 x + 2 y) .$ $- - - - - - - - -$
	Answered by mr W last updated on 24/Dec/21

	$t h e r e a r e m a n y w a y s t o s o l v e .$ $s e e Q 161626 .$ $t h e m e t h o d i p r e f e r :$ $x^{2} + 9 y^{2} + 4 x + 18 y - 23 = 0$ ${(x + 2)}^{2} + 9 {(y + 1)}^{2} = 36$ ${(\frac{x + 2}{6})}^{2} + {(\frac{y + 1}{2})}^{2} = 1$ $l i n e 3 x + 2 y = k s h o u l d t a n g e n t e l l i p s e .$ $6^{2} \times 3^{2} + 2^{2} \times 2^{2} = {(3 \times (- 2) + 2 \times (- 1) - k)}^{2}$ $6^{2} \times 3^{2} + 2^{2} \times 2^{2} = {(k + 8)}^{2}$ $k = - 8 \pm 2 \sqrt{85}$ $\Rightarrow {(3 x + 2 y)}_{m i n} = - 8 - 2 \sqrt{85}$ $\Rightarrow {(3 x + 2 y)}_{m a x} = - 8 + 2 \sqrt{85}$
	Commented by mnjuly1970 last updated on 24/Dec/21

	$t h a n k s a l o t s i r W .$
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