Question

Let C: y² = 4ax, a>0 be a parabola with Focus F. Through point P on parabola in first Quadrant and Q on negative x-axis the tangent PQ is drawn to C with |PQ|=2. Circles C₁ and C₂ are tangent to line OP at P and are both tanget to x-axis. The coordinate of F for which sum of areas of C₁ and C₂ is minimum is. $\left(\frac{1}{\sqrt{n}-\sqrt{n}}, 0\right)$. Then n is ___

Answer 1

The provided problem statement contains a typo in the coordinate of F. The expression $\left(\frac{1}{\sqrt{n}-\sqrt{n}}, 0\right)$ simplifies to $\left(\frac{1}{0}, 0\right)$, which is undefined. Assuming there is a typo and the intended coordinate of F is of the form $(a,0)$ where $a$ is related to $n$, and that the sum of areas is minimized for a specific value of $a$. The sum of areas $A = 4\pi \frac{3+\sqrt{1+4/a^2}}{1+\sqrt{1+4/a^2}}$. This area is minimized when $a$ is minimized. If we assume the question implies a specific value of $a$ is obtained, and if the coordinate of F was intended to be $\left(\frac{1}{k}, 0\right)$ for some $k$, then $a = \frac{1}{k}$. The area is minimized as $a \to 0^+$. If the question intended $a = \frac{1}{2\sqrt{n}}$, and the minimum area occurs at this $a$, then we need more information or a corrected problem statement to determine $n$. Given the ambiguity, a definitive answer for $n$ cannot be provided.

Answer 2

The problem statement has a significant typo in the coordinate of the Focus F: $\left(\frac{1}{\sqrt{n}-\sqrt{n}}, 0\right)$. This simplifies to $\left(\frac{1}{0}, 0\right)$, which is an undefined coordinate. This makes it impossible to proceed with finding a specific value for $n$.

Assuming the question meant that the coordinate of F is $(a,0)$ and the sum of areas is minimized for a particular value of $a$ which is related to $n$. The sum of the areas of the two circles is given by $A = 4\pi \frac{3+\sqrt{1+4/a^2}}{1+\sqrt{1+4/a^2}}$. Let $Y = \sqrt{1+4/a^2}$. Then $A = 4\pi \frac{3+Y}{1+Y} = 4\pi \left(1 + \frac{2}{1+Y}\right)$. To minimize $A$, we need to maximize $1+Y$, which means maximizing $Y = \sqrt{1+4/a^2}$. Maximizing $Y$ implies maximizing $1+4/a^2$, which means maximizing $4/a^2$, and thus minimizing $a^2$. Since $a>0$, the sum of areas is minimized as $a \to 0^+$.

If we assume the intended form of F was $\left(\frac{1}{k}, 0\right)$ and this is the coordinate where the minimum occurs, then $a = \frac{1}{k}$. The minimum area occurs as $a \to 0$, implying $k \to \infty$.

If we were to assume a typo like $a = \frac{1}{2\sqrt{n}}$ is the value of $a$ for which the minimum occurs, then $a \to 0$ implies $n \to \infty$. This is unlikely for a typical problem.

Without a corrected statement for the coordinate of F, it is impossible to determine the value of $n$. The question, as stated, is unsolvable.