Question

For what values of $k,$ the system of equations will represent the coincident lines $x+5y-7=0$ and $4x+20y+k=0?$
$\left( a \right) 28$
$\left( b \right) -28$
$\left( c \right)-26$
$\left( d \right)$ None of these

Answer 1

Hint: We know that the lines which coincide or lie on top of each other are called the coincident lines. So, the equations of these lines are related. The equation of a line coincident with another is the multiple of the equation of the former.

Complete step-by-step answer:
Let us consider the given equations of lines, $x+5y-7=0$ and $4x+20y+k=0.$
We are asked to find the value of $k$ for which these two lines represent the coincident lines.
We know that the lines which coincide or lie on top of each other are called coincident lines.
So, a line $l$ is said to be coincident with another line $m,$ if the both lie on top of each other in the plane. So, we can say that the equation of $m$ is a multiple of the equation of the line $l.$ Or in other words, we can say that the equation of the line $l$ is a multiple of the equation of the line $m.$
Here, the equations are $x+5y-7=0$ and $4x+20y+k=0.$
So, we can say that if these two lines are coincident lines, then these two equations are multiples of each other.
Now, we can see the coefficient of $x$ in the first equation is $1$ and that of the second equation is $4$ which is a multiple of $1.$ And $4=4\times 1.$
Similarly, the coefficient of $y$ in the first equation is $5$ and that of the second equation is $20$ which is a multiple of $5.$ And $20=4\times 5.$
Now, we can see that the constant term $k$ has to be a multiple of the constant term $-7$ in the first equation. And as all other coefficients of the second equations are multiples of $4,$ the constant terms will also be a multiple of $4.$ We can say that the constant term is a multiple of both $4$ and $-7.$ Therefore, $k=4\times -7=-28.$
So, the second equation will become \[~4x+20y-28=0.\]
When we take $4$ out, we will get $4\left( x+5y-7 \right)=4\times 0.$
Hence the given equations represent the coincident lines when $k=-28.$

Note: We know that the parallel lines never have common points because they never intersect with each other. But all the points of coincident lines are common for they lie on top of each other.