Question

If $\left|\begin{array}{ccc}a& b+c& a^2\\ b& c+a& b^2\\ c& a+b& c^2\end{array}\right|=0$, where a, b, c are distinct real numbers, then the straight line ax + by + c = 0 passes through the fixed point

• A. (1, -1)
• B. (-1, -1)
• C. (0, 1)
• D. (1, 1)

Answer 1

The correct option is D

(1, 1)

$\Delta =(a+b+c)\times \left|\begin{array}{ccc}1& b+c& a^2\\ 1& c+a& b^2\\ 1& a+b& c^2\end{array}\right|$
(Applying $C_1\to C_1+C_2$ and taking (a + b + c) common)
$=(a+b+c)\left|\begin{array}{ccc}1& b+c& a^2\\ 0& a-b& (b-a)(b+a)\\ 0& a-c& (c-a)(c+a)\end{array}\right|(R_2\to R_2-R_1,R_3\to R_3-R_1)$
$=(a+b+c)(a-b)(c-a)\left|\begin{array}{ccc}1& b+c& a^2\\ 0& 1& -(b+a)\\ 0& -1& (c-a)\end{array}\right|$
= (a + b + c) (a - b) (c - a) [c + a - b -a]
= -(a + b + c) (a - b) (b - c) (c - a)

Since a, b, c are distinct real numbers then $\Delta =0$ only when a + b + c = 0.
Hence line ax + by + c = 0 passes through the fixed point (1, 1)