Let the abscissae of the two points P and Q on a circle be the roots of x2−4x−6=0 and the ordinates of \[\mathrm{P}\] and \[\mathrm{Q}\] be the roots of y2+2y−7=0. If \[\mathrm{PQ}\] is a diameter of the circle x2+y2+2ax+2by+c=0, then the value of (a+b−c) is \_\_\_\_\_\_\_\_\_\_\_\_\_.
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Official solution
Abscissae of PQ are roots of x2−4x−6=0
Ordinates of PQ are roots of y2+2y−7=0
and PQ is diameter
\[\Rightarrow\] Equation of circle is
x2+y2−4x+2y−13=0
But, given x2+y2+2ax+2by+c=0
By comparison a=−2,b=1,c=−13
\[ \Rightarrow a + b - c = - 2 + 1 + 13 = 12\]