Question

If l, m, n are the sides of $\Delta$LMN and
$\left|\begin{array}{ccc}{l}^2& m^2& n^2\\ (l+1^2)& {(m+1)}^2& {(n+1)}^2\\ {(l-1)}^2& {(m-1)}^2& {(n-1)}^2\end{array}\right|=0$, then the$\Delta$LMN is an

• A. Equilateral
• B. Isosceles
• C. Right angled
• D. None of these

Answer 1

The correct option is B Isosceles

$\left|\begin{array}{ccc}{l}^2& m^2& n^2\\ (l+1{)}^2& {(m+1)}^2& {(n+1)}^2\\ {(l-1)}^2& {(m-1)}^2& {(n-1)}^2\end{array}\right|=0$
$R_2\to R_2-R_3$
$\left|\begin{array}{ccc}{l}^2& m^2& n^2\\ 4l& 4m& 4n\\ {(l-1)}^2& {(m-1)}^2& {(n-1)}^2\end{array}\right|=0$
$C_1\to C_1-C_2,C_2\to C_2-C_3$
$\left|\begin{array}{ccc}(l-m)(l+m)& (m-n)(m+n)& n^2\\ (l-m)& m-n& n\\ (l-m)(l+m)-2(l-m)& (m-n)(m+n)-2(m-n)& {(n-1)}^2\end{array}\right|=0$
$(l-m)(m-n)\left|\begin{array}{ccc}(l+m)& (m+n)& n^2\\ 1& 1& n\\ (l+m-2)& (m+n-2)& {(n-1)}^2\end{array}\right|=0$
$C_1\to C_1-C_2$
$\Rightarrow (l-m)(m-n)\left|\begin{array}{ccc}(l-n)& (m+n)& n^2\\ 0& 1& n\\ (l-n)& (m+n-2)& {(n-1)}^2\end{array}\right|=0$
$R_1\to R_1-R_3$
$(l-m)(m-n)(l-n)\left|\begin{array}{ccc}0& 2& 2n-1\\ 0& 1& n\\ 1& (m+n-2)& {(n-1)}^2\end{array}\right|=0$
$\Rightarrow (l-m)(m-n)(l-n)=0$
$\Rightarrow l=morm=norl=n$
Hence, the triangle is isosceles