Two circles having centers at $$A(x_1, y_1)$$, $$B(x_2, y_2)$$ having radii 5, 3 respectively are touching each other at a point $$P(x_3, y_3)$$. Then $$\left|\sum (x_1 y_2 - x_2 y_1)\right| =$$

Solution

The cyclic sum $$\left|\sum (x_1 y_2 - x_2 y_1)\right|$$ over the points $$A$$, $$B$$, $$P$$ equals twice the area of $$\triangle ABP$$. When two circles touch, the point of contact lies on the line joining the centres, so $$A$$, $$B$$, $$P$$ are collinear. The area is 0, so the value is 0.

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