If two circles <span class="math-tex">\(x^2 + y^2 +2gx + 2fy = 0\)</span> and <span class="math-tex">\(x^2 + y^2 +2g's + 2f'y = 0\)</span> touch each other then which of the following is true?
Both circles pass through the origin (there is no constant term). If they touch, they must touch at the origin and share the same tangent line there.
The tangent at the origin to \(x^2 + y^2 + 2gx + 2fy = 0\) is \(gx + fy = 0\). For the other circle it is \(g'x + f'y = 0\).
Same line means \(\frac{g}{g'} = \frac{f}{f'}\), that is, \(g'f = gf'\).
Answer: (B)
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