If $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ intersects the x-axis at (a,0) and (b,0) ,then $\frac{a + b}{2} = - g$ .

True

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False

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Solution

The correct option is A
True

we are given the circle intersects the x-axis at (a,0) and (b,0) .Let those points be A and B.we want to

see if $\frac{a + b}{2} = - g \frac{a + b}{2}$ is the x-coordinate of midpoint of A and B.

And one more point to note is -g is the x-coordinate of the centre of the circle.

From the figure we can easily prove that the triangles ACM and BCM are congruent triangles.

$\Rightarrow A M = B M$

$\Rightarrow M = (\frac{a + b}{2}, 0)$ - - - - - - (1)

we also note that the x-coordinate of M and C are same.

$\Rightarrow M = (- g, 0)$ - - - - - - - (2)

From(1) and (2) ,we get $\frac{a + b}{2} = - g$

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If x2 + y2 + 2gx + 2fy + c=0 intersects the x-axis at (a,0) and (b,0) ,then a+b2 = −g.

If $x^{2} + y^{2} + 2 g x + 2 f y + c = 0$ intersects the x-axis at (a,0) and (b,0) ,then $\frac{a + b}{2} = - g$ .