Question
If the roots of $${a_1}{x^2} + {b_1}x + {c_1} = 0$$ are $${\alpha _1},{\beta _1},$$ and those of $${a_2}{x^2} + {b_2}x + {c_2} = 0$$ are $${\alpha _2},{\beta _2}$$ such that $${\alpha _1}{\alpha _2} = {\beta _1}{\beta _2} = 1$$ then
A.
$$\frac{{{a_1}}}{{{a_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{c_2}}}$$
B.
$$\frac{{{a_1}}}{{{c_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{a_2}}}$$
C.
$${a_1}{a_2} = {b_1}{b_2} = {c_1}{c_2}$$
D.
None of these
Answer :
$$\frac{{{a_1}}}{{{c_2}}} = \frac{{{b_1}}}{{{b_2}}} = \frac{{{c_1}}}{{{a_2}}}$$
Solution :
Roots of the second equation are reciprocal of those of the first.
$$\therefore \,\,{c_1}{x^2} + {b_1}x + {a_1} = 0\,\,{\text{and }}{a_2}{x^2} + {b_2}x + {c_2} = 0$$ have both roots common.