The maximum area of a circle centered at the | vedclass

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(B) Let the equation of the circle be $x^2 + y^2 = r^2$. The parabola is given by $y = x^2 - 100$,so $x^2 = y + 100$.Substituting $x^2$ into the circl

Vedclass · Accepted Answer

$403$

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(B) Let the equation of the circle be $x^2 + y^2 = r^2$. The parabola is given by $y = x^2 - 100$,so $x^2 = y + 100$.Substituting $x^2$ into the circle equation: $(y + 100) + y^2 = r^2$,which simplifies to $y^2 + y + (100 - r^2) = 0$.For the circle to be inscribed in the parabola,the circle must be tangent to the parabola. This occurs when the quadratic equation in $y$ has exactly one solution,meaning its discriminant $D = 0$.$D = b^2 - 4ac = 1^2 - 4(1)(100 - r^2) = 0$.$1 - 400 + 4r^2 = 0 \implies 4r^2 = 399 \implies r^2 = \frac{399}{4}$.The area of the circle is $\pi r^2 = \frac{399\pi}{4}$.Given the area is $\frac{a\pi}{b}$ with $a = 399$ and $b = 4$,where $a$ and $b$ are coprime.The value of $a + b = 399 + 4 = 403$.

The maximum area of a circle centered at the origin,which is inscribed in the parabola $y = x^2 - 100$,can be expressed as $\frac{a\pi}{b}$,where $a$ and $b$ are coprime numbers. Then the value of $a + b$ is:

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