The equation $y^2-x^2+2x-1=0$ represents

The correct option is A

A pair of straight lines

The explanation for the correct option:

Determine the curve of the given equation.

An equation of pair of straight lines is given:

$y^2-x^2+2x-1=0...1$

Since the general equation of pair of straight lines is:

$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0...2$

On comparing equation $1$ and equation $2$ we get,

$a=-1,h=0,b=1,g=1,f=0$ and $c=-1$.

Since the Condition for pair of straight lines is:

$abc+2fgh-a{f}^2-b{g}^2-c{h}^2=0$

Substitute the obtained values as follows:

$$\begin{gathered} (-1)1(-1)+2\left(0\right)\left(1\right)\left(0\right)-(-1){\left(0\right)}^2-\left(1\right){\left(1\right)}^2-(-1){\left(0\right)}^2=0 \\ \Rightarrow 1+0+0-1-0=0 \\ \Rightarrow 0=0 \end{gathered}$$

So, the given equation represents a pair of straight lines.

Hence, option $A$ is correct .