Question

If the circle x2+y2+2gx+2fy+c=0 touches x-axis at (x1​,0) then x1​ is the repeated root of?​

Answer 1

Given,

The circle x² + y² + 2gx + 2fy + c = 0 touches x-axis at (x1, 0).

To find,

x1 is the repeated root of ?

Here, x² + y² + 2gx + 2fy + c = 0 is the equation of circle where centre of circle is (-g, -f) and radius of circle is √(g² + f² - c).

if given circle touches x - axis at a point P.

it means at that point (i.e., P), variable y = 0

i.e., x² + (0)² + 2gx + 2f(0) + c = 0

⇒x² + 2gx + c = 0

therefore, point (x1, 0) is the repeated root of equation x² + 2gx + c = 0.

[note : you know very well, roots of quadratic equation are same only when D = b² - 4ac = 0

so, above equation x² + 2gx + c = 0, gives same root only when D = (2g)² - 4c = 0 ⇒g² = c ]

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