Question

If the sum of slopes of the pair of lines represented by $4{{x}^{2}}+2hxy-7{{y}^{2}}=0$ is equal to the product of the slopes, then the value of $h$ is
A. $-6$
B. $-2$
C. $-4$
D. $4$

Answer 1

Hint: In this question, we are to find the equation formed by the gradients of the given two lines. For this, we need to use the product and sum of the slopes of a pair of straight lines. By using them, we get the required relation from the given pair of straight lines equation.



Formula Used:The combined equation of 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 $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$ represents a pair of lines, then the sum of the slopes of the lines is $\dfrac{-2h}{b}$ and the product of the slopes of the lines is $\dfrac{a}{b}$.



Complete step by step solution:Given equation is
$4{{x}^{2}}+2hxy-7{{y}^{2}}=0$
On comparing with the equation of pair of straight lines $a{{x}^{2}}+2hxy+b{{y}^{2}}=0$, we get
$a=4;b=-7$
Consider the gradients of these two lines as ${{m}_{1}}$ and ${{m}_{2}}$.
The sum of these two gradients is
${{m}_{1}}+{{m}_{2}}=\dfrac{2h}{b}=\dfrac{2h}{7}\text{ }...(1)$
And the product of these two gradients is
$\begin{align}
  & {{m}_{1}}{{m}_{2}}=\dfrac{a}{b} \\
 & \Rightarrow {{m}_{1}}{{m}_{2}}=\dfrac{4}{-7}\text{ }...(2) \\
\end{align}$
But it is given that, the sum of slopes is equal to the product of the slopes.
So, by equating (1) and (2), we get
$\begin{align}
  & {{m}_{1}}+{{m}_{2}}={{m}_{1}}{{m}_{2}} \\
 & \Rightarrow \dfrac{2h}{7}=\dfrac{-4}{7} \\
 & \therefore h=-2 \\
\end{align}$
Thus, the value is $h=-2$.



Option ‘C’ is correct



Note: Here to find the required equation that relates the coefficients of the equation of pair of straight lines, we need to use the sum and product of the gradients of these two straight lines. By applying these sum and product formulae, we can calculate the required value of the variable.