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 \[a{x^2}\; + {\text{ }}2hxy{\text{ }} + {\text{ }}b{y^2}\; + {\text{ }}2gx{\text{ }} + {\text{ }}2fy{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0\] 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 \[a{x^2}\; + {\text{ }}2hxy{\text{ }} + {\text{ }}b{y^2}\; + {\text{ }}2gx{\text{ }} + {\text{ }}2fy{\text{ }} + {\text{ }}c{\text{ }} = {\text{ }}0\] , 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 ( \[\Delta \] ) of the general equation then we will have conditions .
Determinant of the general equation will be \[\Delta = \left( {\begin{array}{*{20}{c}}
  {{a_{}}}&h&g \\
  h&b&f \\
  g&f&c
\end{array}} \right)\] , if \[\Delta \] the will become \[0\] . Then , it is a pair of straight lines . Otherwise , if \[\Delta \] is not \[0\] ( \[\Delta \ne 0\] ) . 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 \[\Delta = \left( {\begin{array}{*{20}{c}}
  { - {1_{}}}&0&1 \\
  0&1&0 \\
  1&0&{ - 1}
\end{array}} \right)\] , on solving the determinant , we get
\[{\text{ = }} - 1( - 1 - 0) - 0 + 1(0 - 1)\]
\[{\text{ = }} - 1( - 1) - 0 + 1( - 1)\] , on solving we get
\[ = {\text{ }}1 - 1\]
\[ = {\text{ }}0\] .
 Therefore , we get the \[\Delta \] as \[\Delta = 0\]. 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 ( \[\Delta \] ) from the general equation is obtained using the equation \[abc{\text{ }} + {\text{ }}2fgh{\text{ }}-{\text{ }}a{f^2}\;-{\text{ }}a{f^2}\;-{\text{ }}b{g^2}\;-{\text{ }}c{h^2}\; = {\text{ }}0\].