Question

The lines $a^2{x}^2+bc{y}^2=a(b+c)xy$ will be coincident, if

• A. $a=0$ or $b=c$
• B. $a=b$ or $a=c$
• C. $c=0$ or $a=b$
• D. $a=b+c$

Answer 1

The correct option is A

$a=0$ or $b=c$

Find the condition for coincident lines

Given, $a^2{x}^2+bc{y}^2=a(b+c)xy$

$\Rightarrow a^2{x}^2+bc{y}^2-a(b+c)xy=0$

Comparing the above equation with a homogeneous equation $A{x}^2+2Hxy+B{y}^2=0$, we get

$$\begin{gathered} A=a^2 \\ B=bc \\ H=-\frac{a}{2}(b+c) \end{gathered}$$

For coincident lines, $H^2–AB=0$

$$\begin{gathered} \Rightarrow \frac{a^2{(b+c)}^2}{4}-a^{2}bc=0 \\ \Rightarrow a^2{(b-c)}^2=0 \end{gathered}$$

$\Rightarrow a=0$ and $b=c$

Hence, the correct answer is $A$, $a=0$ or $b=c$ .