Question 1195159
k(x^2+y^2 )+x^2-y^2+x+y=0

kx^2+ky^2+x^2-y^2+x+y=0

(k+1)x^2+(k-1)y^2+x+y=0

Check the expression
{{{-4(k+1)(k-1)}}}
-
{{{-4(k^2-1)}}}, and work with that.  You can find reference information in your book in some online sources.
https://www.brainkart.com/article/Conic-Sections_39168/
https://www.varsitytutors.com/hotmath/hotmath_help/topics/conic-sections-and-standard-forms-of-equations
http://mathcentral.uregina.ca/QQ/database/QQ.09.06/h/robin2.html



For intersecting lines,
{{{k+1=-(k-1)}}}
{{{k+1=-k+1}}}
{{{2k=0}}}
{{{highlight_green(k=0)}}}