Question

The equation $$y^2-x^2+2x-1=0$$ represents ?
A. A pair of straight lines
B. A Circle
C. A Parabola
D. Ellipse

Answer 1

Hint: For the questions related to the conic section always compare the given equation with the general equation of conic section which is $$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$$ if the specified conic section ( circle , parabola ) etc. is not given .

Complete step by step answer:
Given : $$y^2-x^2+2x-1=0$$
On comparing the given equation with the general equation $$a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0$$ , we get
$$a=-1$$ , $$b=1$$ , $$g=1$$ , $$h=0$$ , $$f=0$$ , $$c=-1$$ . Now , for different conic sections we have different conditions . For that we have to find the determinant ( $$\mathrm{\Delta }$$ ) of the general equation then we will have conditions .
Determinant of the general equation will be $$\mathrm{\Delta }=\left(\begin{array}{ccc}{a}_{}& h& g\\ h& b& f\\ g& f& c\end{array}\right)$$ , if $$\mathrm{\Delta }$$ the will become $$0$$ . Then , it is a pair of straight lines . Otherwise , if $$\mathrm{\Delta }$$ is not $$0$$ ( $$\mathrm{\Delta }\ne 0$$ ) . Then , we have
If $$h^2=ab$$ , then it is a parabola .
If $$h^2>ab$$ , then it is a hyperbola .
If $$h^2=0,a=b$$ , then it is a circle .
If $$h^2>ab$$ , then it is an ellipse .
Putting the values of the general equations co – efficients , we get $$\mathrm{\Delta }=\left(\begin{array}{ccc}-1_{}& 0& 1\\ 0& 1& 0\\ 1& 0& -1\end{array}\right)$$ , on solving the determinant , we get
$$\text{ = }-1(-1-0)-0+1(0-1)$$
$$\text{ = }-1(-1)-0+1(-1)$$ , on solving we get
$$=1-1$$
$$=0$$ .
 Therefore , we get the $$\mathrm{\Delta }$$ as $$\mathrm{\Delta }=0$$. So , the given equation represents a pair of straight lines .

So, the correct answer is “Option A”.

Note: The general equation for the conic sections for the is a second degree homogeneous equation where $$a$$ , $$h$$ , $$b$$ does not vary simultaneously . The determinant ( $$\mathrm{\Delta }$$ ) from the general equation is obtained using the equation $$abc+2fgh-a{f}^2-a{f}^2-b{g}^2-c{h}^2=0$$.