Question

Determine whether the lines are parallel, intersect, or coincide. x-5y=0, y+1=1/5(x+5)

Answer 1

The lines are two coinciding lines or the same lines.

Solution-

The given line equations are

the first one,

`\Rightarrow x-5y=0`

the second one,

`\Rightarrow y+1=(1)/(5)(x+5)\\\\\Rightarrow 5y+5=x+5\\\\\Rightarrow 5y=x\\\\\Rightarrow x-5y=0`

As we know two line equations
`A_1x+B_1y+C_1=0` and
`A_2x+B_2y+C_2=0` will be,

1. Parallel if,
   `(A_1)/(A_2)=(B_1)/(B_2)`
2. Coincide if,
   `(A_1)/(A_2)=(B_1)/(B_2)=(C_1)/(C_2)`
3. Intersect if,
   `(A_1)/(A_2)\\eq (B_1)/(B_2)`

As here,

`\Rightarrow (A_1)/(A_2)=(B_1)/(B_2)=(C_1)/(C_2)`

`\Rightarrow (1)/(1)=(-5)/(-5)`

(C₁ and C₂ aren't considered as they are 0)

Therefore, the lines are two coinciding lines or the same lines.