JEE MainMathematicsModerateCircles

Find the number of integer values of k for which the equation
$x^{2} + y^{2} + (k - 1)x - ky + 5 = 0$ represents a circle whose radius
cannot exceed 3.
k ds mu iw.kk±dh; ekuksa Kkr dhft, ftuds fy,
lehdj.k $x^{2} + y^{2} + (k - 1)x - ky + 5 = 0$ ,d,slk o`Ùk fu:fir djrk

Show answer and solution

x^{2} + y^{2} + (k - 1)x - ky + 5 = 0

this represents a circle whose radius R is

\sqrt{\left( \frac{k - 1}{2} \right)^{2} + \left( \frac{k}{2} \right)^{2} - 5}

\sqrt{\left( \frac{k - 1}{2} \right)^{2} + \left( \frac{k}{2} \right)^{2} - 5} \leq 3\frac{k^{2} - 2k + 1}{4} + \frac{k^{2}}{4} - 5 \leq 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ 0 \leq 2k^{2} - 2k - 19 \leq 36

\Rightarrow k \in \left\lbrack \frac{1 - \sqrt{111}}{2},\frac{1 - \sqrt{37}}{2} \right\rbrack \cup \left\lbrack \frac{1 + \sqrt{37}}{2},\frac{1 + \sqrt{111}}{2} \right\rbrack

\because

k is an integer

\therefore

k ∈ number of
possible integral values of k is

Source: Grass-G / M-27 Mathematics Paper 2 15 Jan.docx

Need a video for this idea?

Scan any doubt from your notes and get a whiteboard explanation.