Let the system of linear equations

x + y + $\alpha z=2$

3x + y + z = 4

x + 2z = 1

have a unique solution $\left(x^{\ast },y^{\ast },z^{\ast }\right).$ If $\left(\alpha ,x^{\ast }\right),\left(y^{\ast },\alpha \right)and\left(x^{\ast },-y^{\ast }\right)$ are collinear points, then the sum of absolute values of all possible values of $\alpha$ is

<p>x + y + az = 2………… (i)</p><p>3x + y + z = 4 ………… (ii)</p><p>x + 2z = 1 ……………. (iii)</p><p>x = 1, y = 1, z = 0</p><p> (for unique solution <span contenteditable="false"> <math> <mrow> <mi>a</mi> <mo>≠</mo> <mo>−</mo> <mn>3</mn> </mrow> </math> </span>&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1025" DrawAspect="Content" ObjectID="_1814769114"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;&nbsp;</p><p><span contenteditable="false"> <math> <mrow> <mrow> <mo> (</mo> <mrow> <mi>α</mi> <mo>, </mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> <mo>, </mo> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>, </mo> <mi>α</mi> </mrow> <mo>)</mo> </mrow> <mi>a</mi> <mi>n</mi> <mi>d</mi> <mtext> </mtext> <mtext> </mtext> <mrow> <mo> (</mo> <mrow> <mn>1</mn> <mo>, </mo> <mo>−</mo> <mn>1</mn> </mrow> <mo>)</mo> </mrow> </mrow> </math> </span>are collinear.</p><p><span contenteditable="false"> <math> <mrow> <mfrac> <mrow> <mi>α</mi> <mo>−</mo> <mn>1</mn> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> </mfrac> <mo>=</mo> <mfrac> <mrow> <mo>−</mo> <mn>1</mn> <mo>−</mo> <mi>α</mi> </mrow> <mrow> <mn>1</mn> <mo>−</mo> <mn>1</mn> </mrow> </mfrac> <mo>⇒</mo> <mn>1</mn> <mo>−</mo> <msup> <mrow> <mi>α</mi> </mrow> <mrow> <mn>2</mn> </mrow> </msup> <mo>=</mo> <mn>0</mn> <mo>⇒</mo> <mi>α</mi> <mo>=</mo> <mo>±</mo> <mn>1</mn> </mrow> </math> </span> &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp; &nbsp;&lt;!- [if gte mso 9]>&lt;xml&gt; <o:OLEObject Type="Embed" ProgID="Equation.DSMT4" ShapeID="_x0000_i1027" DrawAspect="Content" ObjectID="_1814769116"> </o:OLEObject>&lt;/xml&gt;&lt;! [endif]-&gt;&nbsp;</p><p>Sum of absolute value = 2</p>

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