Question 1184257

Show that the line {{{x + y = q}}} will intersect the curve {{{x^2 - 2x + 2y^2 = 3}}} in two distinct points if {{{q^2 < 2q + 5}}}



{{{ x + y = q }}}.......eq.1
{{{x^2 - 2x + 2y^2 = 3}}}..........eq.2
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{{{  y = q-x }}}.......eq.1

{{{x^2 - 2x + 2y^2 = 3}}}..........eq.2...substitute {{{y}}}

{{{x^2 - 2x + 2(q-x )^2 = 3}}}

{{{x^2 - 2x + 2q^2 - 4qx + 2x^2= 3}}}

{{{3x^2 - 2x- 4qx + 2q^2  - 3=0}}}

{{{3x^2 - (2+ 4q)x + (2q^2  - 3)=0}}}


use discriminant
the line will intersect  the curve in {{{two}}}{{{ distinct}}}{{{ points}}} if

{{{highlight(b^2-4ac >0)}}}  

in your case {{{a=3}}}, {{{b=- (4q+2)}}}, {{{c=(2q^2  - 3)}}}

{{{(- (4q+2))^2-4*3(2q^2  - 3) >0}}}

{{{16q^2 + 16q + 4-24q^2 +36 >0}}}

{{{16q^2 + 16q + 4-24q^2 +36 >0}}}

{{{-8q^2 + 16q + 40 >0}}}........divide by {{{8}}}

{{{-q^2 + 2q + 5 >0}}}

{{{  2q + 5 >q^2}}}

or {{{highlight(q^2<2q + 5)}}}