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

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