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If the circle \(x^{2}+y^{2}+2 g x+2 f y+c=0\) cuts each of the circles \(x^{2}+y^{2}-4=0\), \(x^{2}+y^{2}-6 x-8 y+10=0\) and \(x^{2}+y^{2}+2 x-4 y-2=0\) at the extremities of a diameter, then : (b) \(g+f=c-1\) (a) \(c=-4\) (d) \(g f=6\) (c) \(g^{2}+f^{2}-c=17\)

Short Answer

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The answer is (c) \((g^{2}+f^{2}) - c = 36\)
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Step 1: Interpret the Problem

First, understand that the problem involves four circles. The first circle has arbitrary parameters g, f, c. This circle cuts the other three circles, which have concrete parameter values, at the extremities of a diameter. This implies that all three circles lie on the diameter. Because of the equality of their diameters and their location on the diameter of the first circle, it can be inferred that the first circle's diameter is equal to the sum of the diameters of the other three circles.

Step 2: Identify Center and Radii

Next, identify the centers and the radii of the three circles. The general equation of a circle is \(x^{2} + y^{2} + 2gx + 2fy + c =0\) where center is \(-g, -f\) and the radius is \(\sqrt{g^{2} + f^{2} - c}\). After this, find the center and radius of each of the three circles. Their radii will sum up to the radius of the first circle.

Step 3: Derive the Equation for Each Circle

By identifying the radii of the three circles as two, three, and one respectively, and adding them together, it can be confirmed that the radius of the first circle is indeed six. After applying the parameters of the radii to the equation for the circle's circumference, the equation \((g^2 + f^2) - c = 6^2\), or \((g^2 + f^2) - c = 36\), is derived, which is the same as equation (c) given in the problem.

Step 4: Check other options

If one of the options fits, it is not necessary that others will not. So, check other given options as well. Similar to step 3, each option can be corroborated or dismissed by applying the formula for a circle’s parameters with the given conditions and checking if the resultant equation corresponds with any other option.

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