If the equation $7 x^2-14 x y+p y^2-12 x+q y-4=0$ represents a pair of parallel lines then the value of $\sqrt{p^2+q^2-p q}$ is

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Aicharya · Accepted Answer

Given equation of pair of lines is $\begin{aligned} & 7 x^2-14 x y+p y^2-12 x+q y-4=0 \\ & \therefore \quad \mathrm{a}=7, \mathrm{~b}=\mathrm{p}, \mathrm{c}=-4, \mathrm{~h}=-7, \mathrm{~g}=-6, \mathrm{f}=\frac{\mathrm{q}}{2} \end{aligned}$
The lines are parallel. $\begin{aligned} \therefore \quad & h^2=a b \\ & \Rightarrow(-7)^2=7 p \\ & \Rightarrow p=7 \end{aligned}$
Now, $\mathrm{abc}+2 \mathrm{fgh}-\mathrm{af}^2-\mathrm{bg}^2-\mathrm{ch}^2=0$ $\begin{aligned} \Rightarrow 7(7)(-4)+2\left(\frac{q}{2}\right) & (-6)(-7) \\ & -7\left(\frac{\mathrm{q}}{2}\right)^2-7(-6)^2-(-4)(-7)^2=0 \end{aligned}$ $\begin{aligned} & \quad \Rightarrow-196+42 q-\frac{7 q^2}{4}-252+196=0 \\ & \Rightarrow q^2-24 q+144=0 \\ & \Rightarrow(q-12)^2=0 \\ & \Rightarrow q=12 \\ & \therefore \quad \sqrt{p^2+q^2-p q}=\sqrt{49+144-84}=\sqrt{109}\end{aligned}$

If the equation $7 x^2-14 x y+p y^2-12 x+q y-4=0$ represents a pair of parallel lines then the value of…

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