Question

If the equation $3{{x}^{2}}+xy-{{y}^{2}}-3x+6y+k=0$ represents a pair of lines, then $k$ is equal to
A. $9$
B. $1$
C. $0$
D. $-9$

Answer 1

Hint: In this question, we need to find the value of $k$ in the equation that represents two straight lines. So, we can apply the formula $\Delta =0$ to find the required value.



Formula Used:The equation of the pair of straight lines is written as
$H\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}=0$
This is called a homogenous equation of the second degree in $x$ and $y$
And
$S\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0$
This is called a general equation of the second degree in $x$ and $y$.
If ${{h}^{2}}=ab$, then $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ represents coincident lines.
If ${{h}^{2}}>ab$, then $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ represents two real and different lines that pass through the origin.
Thus, the equation $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ represents two lines. They are:
$ax+hy\pm y\sqrt{{{h}^{2}}-ab}=0$
If $S\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0$ represents a pair of lines, then
i) $abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0$ and
ii) ${{h}^{2}}\ge ab,{{g}^{2}}\ge ac,{{f}^{2}}\ge bc$



Complete step by step solution:Given equation is
$3{{x}^{2}}+xy-{{y}^{2}}-3x+6y+k=0\text{ }...(1)$
But we have the general equation of pair lines as
$S\equiv a{{x}^{2}}+2hxy+b{{y}^{2}}+2gx+2fy+c=0\text{ }...(2)$
Comparing (1) and (2), we get
$a=3;h=\dfrac{1}{2};b=-1;g=\dfrac{-3}{2};f=3;c=k$
If the given equation (1) represents two pairs of lines, then
$\Delta =abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0\text{ }...(3)$
On substituting the above values in (3), we get
$\begin{align}
  & abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0 \\
 & \Rightarrow (3)(-1)(k)+2(3)(\dfrac{-3}{2})(\dfrac{1}{2})-3{{\left( 3 \right)}^{2}}-(-1){{\left( \dfrac{-3}{2} \right)}^{2}}-(k){{\left( \dfrac{1}{2} \right)}^{2}}=0 \\
 & \Rightarrow -3k-\dfrac{9}{2}-27+\dfrac{9}{4}-\dfrac{k}{4}=0 \\
 & \Rightarrow 3k+\dfrac{k}{4}=-\dfrac{9}{2}+\dfrac{9}{4}-27 \\
 & \Rightarrow \dfrac{13k}{4}=\dfrac{-117}{4} \\
 & \therefore k=-9 \\
\end{align}$
Thus, the value is $k=-9$.



Option ‘D’ is correct



Note: Here, the given equation represents pair of lines. So, the given equation should satisfy the condition we have $abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0$. Then, by substituting the values into this condition, we get the required values. In this problem, we need to find the coefficient of ${{x}^{2}}$ in the given equation. So, the we applied above formula. On simplifying, we get the required value.