Question 1085163
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<U>HINT</U>: {{{x^2 - 2xy + y^2}}} = {{{(x-y)^2}}}.


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Therefore,

{{{x^2+y^2-2xy+10x-2y+9}}} = {{{0}}}  is equivalent to

{{{(x-y)^2 + 10x - 2y}}} = {{{-9}}}.   (1)


Now introduce new variables

u = x - y,   (2)
v = x + y.   (3)


From (2) and (3) you have

x = {{{(u + v)/2}}},    (4)
y = {{{(-u + v)/2}}}.   (5)


Now substitute these things into (1). You will have

{{{u^2 + 10((u+v)/2) - 2((v-u)/2)}}} = {{{-9}}},    or, equivalently,

{{{u^2 + 5(u+v) - (v-u)}}} = {{{-9}}},    or, equivalently,

{{{u^2 + 6u + 4v}}} = {{{-9}}}, 

{{{(u+3)^2 + 4v}}} = {{{-9 + 9}}},

{{{(u+3)^2 + 4v}}} = {{{0}}},

{{{(u+3)^2}}} = {{{-4v}}}.


Which conic section is this ??


Correct !  It is a PARABOLA, in coordinates (u,v).
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Solved.