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Algebra Difficulty 4.8 AIME Find the answer

Find the area of the region in the coordinate plane where the discriminant of the quadratic ax2+bxy+cy2=0ax^2 + bxy + cy^2 = 0 is not positive.

A number or a short expression. Spacing and $ signs are ignored.

Solution

To find the region in question, we want to find (a,b)(a, b) such that the discriminant of the quadratic is not positive. In other words, we want 4(a+b−7)2−4(a)(2b)≤0⇔a2+b2−7a−7b+49≤0⇔(a−7)2+(b−7)2≤494(a+b-7)^{2}-4(a)(2b) \leq 0 \Leftrightarrow a^{2}+b^{2}-7a-7b+49 \leq 0 \Leftrightarrow(a-7)^{2}+(b-7)^{2} \leq 49 which is a circle of radius 7 centered at (7,7)(7,7) and hence has area 49π49 \pi.

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