Let the abscissae of the two points P and Q on a circle be the roots of ${\mathrm{x}}^2-4\mathrm{x}-6=0$ and the ordinates of P and Q be the roots of ${\mathrm{y}}^2+2\mathrm{y}-7=0$ If PQ is a diameter of the circle ${\mathrm{x}}^2+{\mathrm{y}}^2+2\mathrm{ax}+2\mathrm{by}+\mathrm{c}=0$, then the value of $\left(\mathrm{a}+\mathrm{b}-\mathrm{c}\right)$ is

Equation of circle diameter form

$\left(\mathrm{x}-{\mathrm{x}}_1\right)\left(\mathrm{x}-{\mathrm{x}}_2\right)+\left(\mathrm{y}-{\mathrm{y}}_1\right)\left(\mathrm{y}-{\mathrm{y}}_2\right)=0$

(where x₁ , x₂ are the roots of ${\mathrm{x}}^2-4\mathrm{x}-6=0$ and y₁, y₂ are the roots of ${\mathrm{y}}^2+2\mathrm{y}-7=0$)

${\mathrm{x}}^2+{\mathrm{y}}^2-4\mathrm{x}+2\mathrm{y}-13=0$

Now,

Compare it with the given equation, we get

$\mathrm{a}=-2, \mathrm{b}=1, \mathrm{c}=-13$

Now

$\mathrm{a}+\mathrm{b}-\mathrm{c}=12$