Question

find the value of k which the pair of linear equations kx+y=k² and x +ky=1 have infinitely many solutions ​

Answer 1

Answer:

k = 1

Step-by-step explanation:

For fixed k,

kx + y = k² represents a line

and

x + ky = 1 represents a line.

There are infinitely many simultaneous solutions only if these two lines have infinitely many points in common, and this means that they must be the same line.  So one equation is just a multiple of the other.  So...

ratio between x coefficients = ratio between y coefficients = ratio between constants

=> k / 1 = 1 / k = k² / 1

=> k² = 1  and  k = k²

=> k = 1

Answer 2

for infinitly many solutions

a1/a2=b1/b2=c1/c2

k/1=1/k=k²/1

taking (i) and (ii)

k/1=1/k

k²=1

k=+1 and -1...............(1)

taking (ii) and (iii)

1/k=k²/1

k³=1

k=1................(2)

from (1) and (2)

k=+1[because +1is common value of k in (1) and (2)]

hence k=+1