Question

Show that the straight lines given by. $(2+k)x+(1+k)y=5+7k$ for different values of k pass through a fixed point. Also find that point.

Answer 1

$(2+k)x+(1+k)y=5+7k$

or$(2x+y-5)+k(x+y-7)=0$

It is of the form $L_1+k{L}_2=0$ i.e., the equation of line passing through the intersection of 2 lines $L_1$ and $L_2$

So, it represents a line passing through

$2x+y-5=0$ and $x+y-7=0$

Sovling the two equation we get $(-2,9).$

Which is the fixed point through which the given line pass.For any value of k.