Question

Show that equation x²+2xy+2y²+2x+2y+1=0 does not represent pair of line​

Answer 1

Explanation:

The general quadratic equation
`ax^2+2hxy+by^2+2gx+2fy+c=0` represents pair of straight lines if its discriminant
`\Delta=0` and
`h^2-ab\geq0.` The discriminant of this equation is given by the determinant,

`\Delta=\left|\begin{array}{ccc}a&f&g\\f&b&h\\g&h&c\end{array}\right|`

Here the given equation is,

`\longrightarrow x^2+2xy+2y^2+2x+2y+1=0`

So,

`a=1`

`b=2`

`c=1`

`f=1`

`g=1`

`h=1`

Thus,

`\longrightarrow\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\1&1&1\end{array}\right|`

Performing the operation
`R_3\to R_3-R_1` over this determinant,

`\longrightarrow\Delta=\left|\begin{array}{ccc}1&1&1\\1&2&1\\0&0&0\end{array}\right|`

Now the 3rd row is completely so the discriminant is equal to zero.

`\longrightarrow\Delta=0`

So our equation represents degenerate conics, which are,

pair of distinct straight lines intersecting each other at any point if
`h^2-ab > 0.`

pair of straight lines, either coincident or distinct but parallel to each other, if
`h^2-ab=0.`

a single point if
`h^2-ab < 0.`

We see that,

`\longrightarrow h^2-ab=1^2-1* 2=-1`

`\longrightarrow h^2-ab < 0`

So our equation represents a single point.

Hence the equation does not represent pair of straight lines.