Question

Equation $3{{x}^{2}}+7xy+2{{y}^{2}}+5x+5y+2=0$ represents two straight lines
A. Pair of straight line
B. Ellipse
C. Hyperbola
D. None of these

Answer 1

Hint: In this question, we need to find the form of the given equation. For this, we need to compare the given equation with the general forms of the lines and the curves mentioned in the question. Since the given equation matches the general form of pair of straight lines, we can apply the formula $\Delta =0$ to verify whether it is the equation of pair of straight lines.

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}}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
$3{{x}^{2}}+7xy+2{{y}^{2}}+5x+5y+2=0\text{ }...(1)$
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 came to know that, they are in the same form. So, we can use the condition of pair of straight lines to prove the given equation represents two pairs of straight lines.
Here on comparing, we get
$a=3;h=\dfrac{7}{2};b=2;g=\dfrac{5}{2};f=\dfrac{5}{2};c=2$
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
$\begin{align}
  & abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}} \\
 & \Rightarrow (3)(2)(2)+2(\dfrac{5}{2})(\dfrac{5}{2})(\dfrac{7}{2})-3{{\left( \dfrac{5}{2} \right)}^{2}}-(2){{\left( \dfrac{5}{2} \right)}^{2}}-(2){{\left( \dfrac{7}{2} \right)}^{2}} \\
 & \Rightarrow 12+\dfrac{175}{4}-\dfrac{75}{4}-\dfrac{25}{2}-\dfrac{49}{2} \\
 & \Rightarrow \Delta =0 \\
\end{align}$
Thus, the given equation represents the equation of pair of straight lines since it satisfies the condition at (3).



Option ‘A’ is correct



Note: Here, the given equation represents pair of lines. For any equation that represents a pair of straight lines, it must satisfy the condition, we have $abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0$. On substituting the values into the L.H.S of this condition, if we get zero on the R.H.S, then that equation represents the pair of straight lines.