First slide

Theory of equations

Question

If the equation $(m - n) x^{2} + (n - l) x + l - m = 0$ has equal roots, then l, m and n satisfy

Moderate

A

$2 l = m + n$
B

$2 m = n + l$
C

$m = n + l$
D

$l = m + n$

Solution

Roots are equal so $b^{2} - 4 ac = 0$
$\Rightarrow {(n - l)}^{2} - 4 (m - n) (l - m) = 0$
$\Rightarrow n^{2} + l^{2} - 2 nl - 4 (ml - nl - m^{2} + mn) = 0$
$\Rightarrow n^{2} + l^{2} - 2 nl - 4 ml + 4 nl + 4 m^{2} - 4 mn = 0$
$\Rightarrow l^{2} + n^{2} + {(2 m)}^{2} + 2 nl - 4 mn - 4 ml = 0$
$\Rightarrow {(l + n - 2 m)}^{2} = 0 \Rightarrow l + n = 2 m \Rightarrow l, m, n are in A . P .$

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