If $sin θ$ and $cos θ$ are the roots of the equation $m x^{2} + n x + 1 = 0$ , then find the relation between m and n.

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Solution

The given equation is $m x^{2} + n x + 1 = 0$
Here, $a = m, b = n, c = 1$
$sin θ + cos θ = \frac{- b}{a} = \frac{- n}{m}$

$sin θ + cos θ = \frac{c}{a} = \frac{1}{m}$

Consider,
$sin θ + cos θ = \frac{- n}{m}$

$(sin θ + cos θ)^{2} = (\frac{- n}{m})^{2}$

$s i n^{2} θ + c o s^{2} θ + 2 sin θ cos θ = \frac{n^{2}}{m^{2}}$

$1 + 2 sin θ cos θ = \frac{n^{2}}{m^{2}}$

$1 + 2 (\frac{1}{m}) = \frac{n^{2}}{m^{2}}$ $[sin θ cos θ = \frac{1}{m}]$

$1 + \frac{2}{m} = \frac{n^{2}}{m^{2}}$

$n^{2} - m^{2} = 2 m$ .

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