If the graphs of y=x^2 +2ax+6b and y=x^2 +2bx+6a intersect at? only one point in the xy−plane , what is the x−coordinate of the point of intersection ?


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Question Number 118868 by bemath last updated on 20/Oct/20

$I f t h e g r a p h s o f y = x^{2} + 2 a x + 6 b$ $a n d y = x^{2} + 2 b x + 6 a i n t e r s e c t a t ?$ $o n l y o n e p o i n t i n t h e x y - p l a n e$ $, w h a t i s t h e x - c o o r d i n a t e o f t h e$ $p o i n t o f i n t e r s e c t i o n ?$
	Answered by bobhans last updated on 20/Oct/20

	$L e t (p, q) b e t h e p o i n t o f i n t e r s e c t i o n o f$ $t h e t w o g r a p h . T h e n q = p^{2} + 2 a p + 6 b = p^{2} + 2 b p + 6 a$ $i t f o l l o w s t h a t 2 (a - b) p = 6 (a - b) .$ $B e c a u s e t h e t w o g r a p h s i n t e r s e c t a t$ $o n l y o n e p o i n t, i t d o e s m e a n 2 p = 6 a n d h e n c e p = 3$ $t h e r e f o r e q = 3^{2} + 6 a + 6 b; q = 9 + 6 a + 6 b$
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