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Circles Part 1
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Section:
Mathematics
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© examsnet.com
Question : 6
Total: 100
Let the equation
x
2
+
y
2
+
p
x
+
(
1
−
p
)
y
+
5
=
0
represent circles of varying radius
r
∈
(
0
,
5
]
. Then, the number of elements in the set
S
=
{
q
:
q
=
p
2
and q is an integer} is
[27 Aug 2021 Shift 1]
Your Answer:
Validate
Solution:
Given equation,
x
2
+
y
2
+
p
x
+
(
1
−
p
)
y
+
5
=
0
Radius
,
r
=
√
(
p
2
)
2
+
(
1
−
p
2
)
2
−
5
=
√
2
p
2
−
2
p
−
19
4
Now, r ∈ [0, 5]
$0
⇒
0
<
√
2
p
2
−
2
p
−
19
4
≤
5
⇒
0
<
2
p
2
−
2
p
−
19
4
≤
25
⇒
2
p
2
−
2
p
−
19
>
0
and
2
p
2
−
2
p
−
19
≤
100
⇒
2
p
2
−
2
p
−
19
>
0
and
2
p
2
−
2
p
−
19
≤
100
⇒
p
2
−
p
−
19
2
>
0
and
2
p
2
−
2
p
−
119
≤
0
⇒
p
∈
(
−
∞
,
1
−
√
39
2
)
‌
⋃
(
1
+
√
39
2
,
∞
)
and
p
∈
[
1
−
√
239
2
,
1
+
√
239
2
]
p
∈
[
1
−
√
239
2
,
1
−
√
39
2
]
‌
⋃
[
1
+
√
39
2
,
1
+
√
239
2
]
⇒
p
2
∈
[
(
1
−
√
39
2
)
2
,
(
1
+
√
239
2
)
2
]
∵
q
=
p
2
,
q
∈
Z
⇒
q
=
7
,
8
,
9
,
10
,
.
.
.
,
67
⇒ 61 integers in S
∴ n (S) = 61
© examsnet.com
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