The figure formed by the pairs of lines $6 x^2+13 x y+6 y^2=0$ and $6 x^2+13 x y+6 y^2+10 x+10 y+4=0$, is a
The figure formed by the pairs of lines $6 x^2+13 x y+6 y^2=0$ and $6 x^2+13 x y+6 y^2+10 x+10 y+4=0$, is a
Square
Parallelogram
Rhombus
Rectangle
Solution
(c) Equation of pair of lines,
$
\begin{aligned}
& 6 x^2+13 x y+6 y^2=0 \\
& 6 x^2+9 x y+4 x y+6 y^2=0 \\
& 3 x(2 x+3 y)+2 y(2 x+3 y)=0 \\
& (3 x+2 y)(2 x+3 y)=0 \\
& \therefore \quad 3 x+2 y=0 \text { and } 2 x+3 y=0
\end{aligned}
$
Similarly, for another equation of pairs of lines
$
6 x^2+13 x y+6 y^2+10 x+10 y+4=0 \text { are } 3 x+2 y+2=0
$
$
\text { and } 2 x+3 y+2=0
$
Here, $A B$ is parallel to $C D$ and $B C$ is parallel to $A D$. $\therefore \quad A B C D$ is a parallelogram.
But also $A B=B C=C D=A D=\frac{2}{\sqrt{13}}$ Hence, $A B C D$ is a rhombus.
Hence, ABCD is a rhombus
