Question

The number of integer values of $k$ for which the equation $x^2+y^2+(k-1)x-ky+5=0$ represent a circle whose radius cannot exceed $3,$ is

• A. 5
• B. 4
• C. 11
• D. 10

Answer 1

The correct option is B

4

Let the radius be $r$

$\Rightarrow r\le 3\Rightarrow 0\ge r^2\le 9$

Now, radius of the given circle is $\sqrt{\frac{1}{4}(k-1{)}^2+\frac{1}{4}{k}^2-5}$

$\Rightarrow 0\le \frac{1}{4}(k-1{)}^2+\frac{1}{4}{k}^2-5\le 9$

$\Rightarrow 0\le k^2-2k+1+k^2-20\le 36$

$\Rightarrow 19\le 2k^2-2k\le 55$

$\Rightarrow 38\le 4k^2-4k\le 110$

$\Rightarrow 39\le 4k^2-4k+1\le 111$

$\Rightarrow 39\le (2k-1{)}^2\le 111$

$\Rightarrow (2k-1{)}^2=49,64,81,100$

$\Rightarrow 2k-1=\pm 7,\pm 9$ [As $k$ is an integer]

$\Rightarrow k$ can take four integral values.