Consider the triangle with the vertices $A (1, 2, 2), B (3, 1, 2) & C (1, 1, 1)$ and the plane $P : x - y + z = 7$
Orthogonally project the given triangle ABC, on to the given plane $x - y + z = 7$ , the vertices A, B & C become $A_{1}, B_{1} & C_{1}$ respectively
Reflect the given triangle ABC, about the given plane $x - y + z = 7$ , the vertices A, B & C become $A_{2}, B_{2} & C_{2}$ respectively
And let $A B = c, B C = a & C A = b$ and $A_{1} B_{1} = c_{1}, B_{1} C_{1} = a_{1} & C_{1} A_{1} = b_{1}$
also $A_{2} B_{2} = c_{2}, B_{2} C_{2} = a_{2} & C_{2} A_{2} = b_{2}$
Then, which of the following is / are true

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Number of real roots of the equation $c_{1} x^{2} + a_{1} x + b_{1} = 0$ is 2

Number of real roots of the equation $c_{2} x^{2} + a_{2} x + b_{2} = 0$ is 0

$B_{1} C_{1} < B C = B_{2} C_{2}$

$A C = A_{1} C_{1} = A_{2} C_{2}$

answer is B, C, D.

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Detailed Solution

$A (1, 2, 2), B (3, 1, 2) & C (1, 1, 1) A_{1} (1, 2, 2), B_{1} (3, 1, 2) & C_{1} (1, 1, 1) A_{2} (1, 2, 2), B_{2} (3, 1, 2) & C_{2} (1, 1, 1) A B = \sqrt{5}, B C = \sqrt{5}, C A = \sqrt{2} A_{1} B_{1} = \sqrt{2}, B_{1} C_{1} = \sqrt{2}, C_{1} A_{1} = \sqrt{2} A_{2} B_{2} = \sqrt{5}, B_{2} C_{2} = \sqrt{5}, C_{2} A_{2} = \sqrt{2}$