For what value of k will the equation #12x^2-10xy+2y^2+11x-5y+k=0# represent a pair of lines?

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Answer 1 · 2018-08-08T19:17:41

#k=2#

We know that ,
#color(blue)("If " ax^2+2hxy+by^2+2gx+2fy+c=0, a^2+h^2+b^2!=0#

#color(blue)("represent a pair of lines",then#

#color(blue)(D=|(a,h,g),(h,b,f),(g,f,c)|=0#

We have,

#12x^2-10xy+2y^2+11x-5y+k=0#

#:.a=12,h=-5,b=2,g=11/2,f=-5/2 and c=k#

#:.D=|(12,-5,11/2),(-5,2,-5/2),(11/2,-5/2,k)|=0#

Expanding we get

#12(2k-25/4)+5(-5k+55/4)+11/2(25/2-11)=0#

#24k-75-25k+275/4+275/4-121/2=0#

#:.-k-75+(275+275-242)/4=0#

#:.-k-75+77=0#

#:.-k+2=0#

#:.k=2#