Question

For what values of $p$, ${{y}^{2}}+xy+p{{x}^{2}}-x-2y=0$ represents two straight lines
A. $2$
B. $\dfrac{1}{3}$
C. $\dfrac{1}{4}$
D. $\dfrac{1}{2}$

Answer 1

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



Formula UsedThe 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
${{y}^{2}}+xy+p{{x}^{2}}-x-2y=0\text{ }...(1)$
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
$a=p;h=\dfrac{1}{2};b=1;g=\dfrac{-1}{2};f=-1;c=0$
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
$\begin{align}
  & abc+2fgh-a{{f}^{2}}-b{{g}^{2}}-c{{h}^{2}}=0 \\
 & \Rightarrow (p)(1)(0)+2(-1)(\dfrac{-1}{2})(\dfrac{1}{2})-p{{\left( -1 \right)}^{2}}-(1){{\left( \dfrac{-1}{2} \right)}^{2}}-(0){{\left( \dfrac{1}{2} \right)}^{2}}=0 \\
 & \Rightarrow 0+\dfrac{1}{2}-p-\dfrac{1}{4}-0=0 \\
 & \therefore p=\dfrac{1}{4} \\
\end{align}$
Thus, the value is $p=\dfrac{1}{4}$.


Option ‘C’ 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.