Let the equation x2 + y2 + 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 = p2 and q is an integer} is __________.
Answer (integer)
61
Solution
$$r = \sqrt {{{{p^2}} \over 4} + {{{{(1 - p)}^2}} \over 4} - 5} = {{\sqrt {2{p^2} - 2p - 19} } \over 2}$$<br><br>Since, $r \in (0,5]$<br><br>So, $0 < 2{p^2} - 2p - 19 \le 100$<br><br>$$ \Rightarrow p \in \left[ {{{1 - \sqrt {239} } \over 2},{{1 - \sqrt {39} } \over 2}} \right) \cup \left( {{{1 + \sqrt {39} } \over 2},{{1 + \sqrt {239} } \over 2}} \right]$$ <br><br>so, number of integral values of p<sup>2</sup> is 61.
About this question
Subject: Mathematics · Chapter: Circles · Topic: Equation of a Circle
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