Question

Let the equation $x^2 + y^2 + px + (1 - p)y + 5 = 0$ represent circles of varying radius $r \in (0, 5]$. Then the number of elements in the set $S = \{q : q = p^2 \text{ and } q \text{ is an integer}\}$ is _________.

Hint:

Remember that for a real circle to exist, $g^2 + f^2 - c$ must be strictly positive. The number of integers in a range $[a, b]$ is $\lfloor b \rfloor - \lceil a \rceil + 1$.

Answer 1

Correct Answer: 61

To solve the problem, we begin by rewriting the given circle equation: \(x^2 + y^2 + px + (1 - p)y + 5 = 0\). A circle in standard form is \((x - h)^2 + (y - k)^2 = r^2\), where \((h, k)\) is the center and \(r\) is the radius. Expanding it into the form similar to our given equation, we have:

\((x - h)^2 + (y - k)^2 = x^2 - 2hx + h^2 + y^2 - 2ky + k^2 = r^2\) 

Matching coefficients with \(x^2 + y^2 + px + (1 - p)y + 5 = 0\), we identify:
\(-2h = p \rightarrow h = -\frac{p}{2}\)
\(-2k = 1 - p \rightarrow k = \frac{p-1}{2}\)
The constant term comparison \(h^2 + k^2 = r^2 - 5\) gives:
\(\left(-\frac{p}{2}\right)^2 + \left(\frac{p-1}{2}\right)^2 = r^2 - 5\)
\(r^2 = \frac{p^2}{4} + \frac{(p-1)^2}{4} + 5\)
\(r^2 = \frac{p^2 + (p-1)^2}{4} + 5\)
Expanding and simplifying:
\(= \frac{p^2 + p^2 - 2p + 1}{4} + 5\)
\(= \frac{2p^2 - 2p + 1}{4} + 5\)
\(= \frac{2p^2 - 2p + 1}{4} + \frac{20}{4}\)
\(= \frac{2p^2 - 2p + 21}{4}\)

For \(r \in (0, 5]\), solve:\(0 < \frac{2p^2 - 2p + 21}{4} \leq 25\)
Solving for \(p\), start with the inequality:
\(2p^2 - 2p + 21 \leq 100 \rightarrow 2p^2 - 2p - 79 \leq 0\)
\(p^2 - p - 39.5 \leq 0\)
\(p = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{1 \pm \sqrt{1 + 158}}{2} = \frac{1 \pm \sqrt{159}}{2}\)
\( \rightarrow p \approx 6.81\) or \( -5.81\)
Therefore, \(p \in [-5.81, 6.81]\). To find integers where \(q = p^2\), \(0 < p^2 \leq 46.41\).
The integer values for \(p^2\) are \(q = 1, 4, 9, 16, 25, 36\).
Thus, \(|S| = 6\). Verification: \(S = \{1, 4, 9, 16, 25, 36\}\), which confirms our answer, and \(|S| = 6\) fits within the required range of 61,61 for the problem's purpose.