Question

The condition of representing the coincident lines by the general quadratic equation $f(x,y)=0$, is

• A. $∆=0\text{and}{h}^2=ab$
• B. $∆=0\text{and}a+b=0$
• C. $∆=0\text{and}{h}^2=ab,g^2=ac,f^2=bc$
• D. $h^2=ab,g^2=ac\text{and}{f}^2=bc$

Answer 1

The correct option is C

$∆=0\text{and}{h}^2=ab,g^2=ac,f^2=bc$

Explanation for the correct option:

Finding the condition:

Given, $\mathrm{f}(\mathrm{x},\mathrm{y})=0$

The general equation is given by ${\mathrm{ax}}^2+2\mathrm{hxy}+{\mathrm{by}}^2+2\mathrm{gx}+2\mathrm{fy}+\mathrm{c}=0$,

The condition for straight lines is Determinant $∆=0$,

$$\begin{gathered} \Rightarrow \left|\begin{array}{ccc}\mathrm{a}& \mathrm{h}& \mathrm{g}\\ \mathrm{b}& \mathrm{h}& \mathrm{f}\\ \mathrm{g}& \mathrm{f}& \mathrm{c}\end{array}\right|=0 \\ \Rightarrow \mathrm{abc}+2\mathrm{fgh}–{\mathrm{af}}^2–{\mathrm{bg}}^2–{\mathrm{ch}}^2=0 \end{gathered}$$

Angle between pair of straight lines, $\mathrm{tan\theta }=2\left(\frac{\sqrt{{\mathrm{h}}^2-\mathrm{ab}}}{(\mathrm{a}+\mathrm{b})}\right)$,when${\mathrm{h}}^2=\mathrm{ab},\tan \mathrm{\theta }=0$.

Therefore, the given lines are parallel.

Distance between parallel lines is given by

$d=2\left(\frac{\sqrt{g^2-ac}}{a(a+b)}\right)\text{or}d=2\left(\frac{\sqrt{f^2-bc}}{b(a+b)}\right)$

For coincident lines, $d=0$,=>${\mathrm{g}}^2=\mathrm{ac}\mathrm{or}{\mathrm{f}}^2=\mathrm{bc}$

So, the condition for representing the coincident lines is $∆=0,{\mathrm{h}}^2=\mathrm{ab},{\mathrm{g}}^2=\mathrm{ac}\mathrm{and}{\mathrm{f}}^2=\mathrm{bc}.$

Hence option (C) is the correct answer.