Question

Let the system of linear equations
x + y + az = 2
3x + y + z = 4
x + 2z = 1
have a unique solution (x*, y*, z*). If (α, x*), (y*, α) and (x*, –y*) are collinear points, then the sum of absolute values of all possible values of α is

• A. 4
• B. 3
• C. 2
• D. 1

Answer 1

The Correct Option is C

Given system of equations

\(x + y + az = 2 \)…(i)

\(3x + y + z = 4\) …(ii)

\(x + 2z = 1\) …(iii)

Solving (i), (ii) and (iii), we get

\(x = 1,\)  \(y = 1\) , \(z = 0\) (and for unique solution \(a ≠–3\))

Now, \((α, 1), (1, α)\) and \((1, –1)\) are collinear

\(∴\) \(\begin{vmatrix}    \alpha&1 & 1\\1&\alpha&1\\1&-1 &1   \end{vmatrix}=0\)

⇒ \(α(α + 1) – 1(0) + 1(–1 – α) = 0\)

\(⇒ α^2 – 1 = 0\)

\(∴ α = ±1\)

\(∴\) Sum of absolute values of \(α = 1 + 1 = 2\)

Hence, the correct option is (C): \(2\)