If thee quadratic equations $x^{2} + (\sqrt{A B + A C}) x + \frac{B C}{4} = 0, x^{2} + (\sqrt{B C + A B}) x + \frac{A C}{4} = 0$ and $x^{2} + (\sqrt{A C + B C}) x + \frac{A B}{4} = 0$ are given, where A, B, C are three distinct points in a plane then, which of the following is/are CORRECT?

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If A, B, C are making a triangle then all equations have real & distinct roots.

If A, B, C are collinear, then all equations have real & equal roots.

If A, B, C are collinear, then two equations have real & distinct, one equation has real & equal roots.

for every position of A, B, C in the plane, all six roots are negative.

answer is A, C, D.

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Detailed Solution

$D_{1} = (A B + A C) - B C; D_{3} = (A C + B C) - A B$
$D_{2} = (B C + A B) - A C;$ for $Δ$ all are +ve (Distinct & real)
If collinear then let $A B + B C = A C$
$D_{3} = 0$ & all others will be +ve
$\begin{array}{l} α + β = - v e \\ α β = + v e \end{array}}$ in each case $\Rightarrow$ both roots are negative