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Let the equation x2 y2 px 1 py 5 0 represent circles of... | null (Mathematics) | DivineJEE

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Mathematics

Exam

Mathematics

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 __________.

Correct Answer:

61

Explanation

r = \sqrt {{{{p^2}} \over 4} + {{{{(1 - p)}^2}} \over 4} - 5} = {{\sqrt {2{p^2} - 2p - 19} } \over 2}

Since,

r \in (0,5]

So,

0 < 2{p^2} - 2p - 19 \le 100

\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]

so, number of integral values of p² is 61.

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