Question 1051725
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find the range of values of k for which kx+y=3 meets x²+y²=5 in two distinct points
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kx + y = 3     (1)
x² + y² = 5    (2)

Express y = 3-kx  from (1) and substitute it into (2). You will get

{{{x^2 + (3-kx)^2}}} = 5.

Simplify:

{{{x^2 + 9 - 6kx + k^2x^2}}} = 5,

{{{(k^2+1)*x^2 -6kx + 9-5}}} = 0,

{{{(k^2+1)*x^2 - 6kx + 4}}} = 0.

Discriminant 

d = {{{b^2 - 4ac}}} = {{{(6k)^2 - 4*(k^2+1)*4}}} = {{{36k^2 - 16*(k^2+1)}}} = {{{36k^2 - 16k^2 - 16}}} = {{{20k^2 - 16}}}.

The condition for  "kx+y=3 meets x²+y²=5 in two distinct points"  is  this inequality  d > 0,  or

{{{20k^2 - 16}}} > 0,   or, which is the same (cancel the factor 4)

{{{5k^2 - 4}}} > 0,  or

|k| > {{{sqrt(4/5)}}} = {{{2/sqrt(5)}}}.


<U>Answer</U>.  kx+y=3 meets x²+y²=5 in two distinct points  if  and only if k < {{{-2/sqrt(5)}}}  OR  k > {{{2/sqrt(5)}}}.
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