JEE MAIN - Mathematics (2022 - 26th July Evening Shift - No. 10)
Let the abscissae of the two points $$P$$ and $$Q$$ on a circle be the roots of $$x^{2}-4 x-6=0$$ and the ordinates of $$\mathrm{P}$$ and $$\mathrm{Q}$$ be the roots of $$y^{2}+2 y-7=0$$. If $$\mathrm{PQ}$$ is a diameter of the circle $$x^{2}+y^{2}+2 a x+2 b y+c=0$$, then the value of $$(a+b-c)$$ is _____________.
12
13
14
16
Explanation
Abscissae of PQ are roots of $${x^2} - 4x - 6 = 0$$
Ordinates of PQ are roots of $${y^2} + 2y - 7 = 0$$
and PQ is diameter
$$\Rightarrow$$ Equation of circle is
$${x^2} + {y^2} - 4x + 2y - 13 = 0$$
But, given $${x^2} + {y^2} + 2ax + 2by + c = 0$$
By comparison $$a = - 2,b = 1,c = - 13$$
$$ \Rightarrow a + b - c = - 2 + 1 + 13 = 12$$
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