10
Question
The number of integer values of k for which the equation x2+y2+(k−1)x−ky+5=0 represent a circle whose radius cannot exceed 3, is
The number of integer values of k for which the equation x2+y2+(k−1)x−ky+5=0 represent a circle whose radius cannot exceed 3, is
5
4
11
10
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Solution
The correct option is B 4
Let the radius be r
⇒r≤3⇒0≥r2≤9
Now, radius of the given circle is √14(k−1)2+14k2−5
⇒0≤14(k−1)2+14k2−5≤9
⇒0≤k2−2k+1+k2−20≤36
⇒19≤2k2−2k≤55
⇒38≤4k2−4k≤110
⇒39≤4k2−4k+1≤111
⇒39≤(2k−1)2≤111
⇒(2k−1)2=49, 64, 81, 100
⇒2k−1=±7, ±9 [As k is an integer]
⇒k can take four integral values.
4
Let the radius be r
⇒r≤3⇒0≥r2≤9
Now, radius of the given circle is √14(k−1)2+14k2−5
⇒0≤14(k−1)2+14k2−5≤9
⇒0≤k2−2k+1+k2−20≤36
⇒19≤2k2−2k≤55
⇒38≤4k2−4k≤110
⇒39≤4k2−4k+1≤111
⇒39≤(2k−1)2≤111
⇒(2k−1)2=49, 64, 81, 100
⇒2k−1=±7, ±9 [As k is an integer]
⇒k can take four integral values.
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