Multiple choice

If the area of an equilateral triangle inscribed in the circle, $x^2 + y^2 + 10x + 12y + c=0$ is $27\sqrt{3}sq.$ units then $c$ is equal to:

  1. $20$
  2. $25$
  3. $13$
  4. $-25$
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B Correct answer
Explanation

The circle is x^2 + y^2 + 10x + 12y + c = 0. Center (-5, -6), radius R = sqrt(25 + 36 - c) = sqrt(61-c). Area of inscribed equilateral triangle = (3 * sqrt(3) / 4) * R^2 = 27 * sqrt(3). So, R^2 = 36. 61 - c = 36 => c = 25.

AI explanation

The area of an equilateral triangle inscribed in a circle of radius r is given by the formula Area = (3*sqrt(3)/4) * r^2. We set the given area equal to this formula: 27*sqrt(3) = (3*sqrt(3)/4) * r^2. Solving for the radius, we divide both sides by 3*sqrt(3), yielding 9 = r^2 / 4, so r^2 = 36 and r = 6. The given circle equation is x^2 + y^2 + 10x + 12y + c = 0. The radius squared is calculated as (-g)^2 + (-f)^2 - c. Here, -g = -5 and -f = -6, so the radius squared is (-5)^2 + (-6)^2 - c = 25 + 36 - c. Setting the radius squared to 36, we get 61 - c = 36. Solving for c gives c = 25. The result is 25.

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