If the two circles x^2+y^2+2 g1 x+2 f1 y=0 and x^2+y^2+2 g2...

If the two circles $x^{2}+y^{2}+2 g_{1} x+2 f_{1} y=0$ and $x^{2}+y^{2}+2 g_{2} x+2 f_{2} y=0$ touch each other, then (a) $\mathrm{f}_{1} \mathrm{f}_{2}=\mathrm{g}_{1} \mathrm{~g}_{2}$ (b) $\mathrm{f}_{1} \mathrm{~g}_{1}=\mathrm{f}_{2} \mathrm{~g}_{2}$ (c) $\mathrm{f}_{1} \mathrm{~g}_{2}=\mathrm{f}_{2} \mathrm{~g}_{1}$ (d) $\mathrm{f}_{1} \mathrm{~g}_{2}{ }^{2}=\mathrm{f}_{2} \mathrm{~g}_{1}{ }^{2}$

If the two circles $x^{2}+y^{2}+2 g_{1} x+2 f_{1} y=0$ and $x^{2}+y^{2}+2 g_{2} x+2 f_{2} y=0$ touch each other, then
(a) $\mathrm{f}_{1} \mathrm{f}_{2}=\mathrm{g}_{1} \mathrm{~g}_{2}$
(b) $\mathrm{f}_{1} \mathrm{~g}_{1}=\mathrm{f}_{2} \mathrm{~g}_{2}$
(c) $\mathrm{f}_{1} \mathrm{~g}_{2}=\mathrm{f}_{2} \mathrm{~g}_{1}$
(d) $\mathrm{f}_{1} \mathrm{~g}_{2}{ }^{2}=\mathrm{f}_{2} \mathrm{~g}_{1}{ }^{2}$

Key Concepts

Standard Equation of a Circle

The standard form of a circle is given by (x - h)² + (y - k)² = r², where (h, k) represents the center and r is the radius. Converting a general equation like x² + y² + 2gx + 2fy = 0 into this form, one finds that the center is (-g, -f) and the radius is the square root of (g² + f²), a step that is crucial for analyzing properties of the circle.

Center and Radius Determination

For any circle represented by the equation x² + y² + 2gx + 2fy + c = 0, completing the square reveals that the center is (-g, -f) and the radius is ?(g² + f² - c). In cases where c is zero, as in the given circles, the radius directly becomes ?(g² + f²), a fact that is essential when comparing distances between circles.

Tangency Condition for Circles

Two circles touch each other (are tangent) if the distance between their centers equals either the sum (for external tangency) or the absolute difference (for internal tangency) of their radii. This geometric condition is used to establish a relationship between the centers and radii, and by extension, between the coefficients in the equations of the circles.

Algebraic Relationships Between Coefficients

When the tangency condition is applied to circles written in general form, it leads to specific equations relating the coefficients of the circles. By equating the distance between centers (derived from the coefficients) to the appropriate sum or difference of the radii, one can deduce proportional relationships among the coefficients, thereby linking the geometric condition to an algebraic criterion.

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