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Section:
Mathematics
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© examsnet.com
Question : 6 of 17
Marks:
+1
,
-0
Let T be the line passing through the points
P
(
−
2
,
7
)
P(-2,7)
P
(
−
2
,
7
)
and
Q
(
2
,
−
5
)
Q(2,-5)
Q
(
2
,
−
5
)
.Let
F
1
F_1
F
1
be the set of all pairs of circles
(
S
1
,
S
2
)
(S_1,S_2)
(
S
1
,
S
2
)
such that T is tangents to
S
1
S_1
S
1
at P and tangent to
S
2
S_2
S
2
at Q, and also such that
S
1
S_1
S
1
and
S
2
S_2
S
2
touch each other at a point, say, M. Let
E
1
E_1
E
1
be the set representing the locus of M as the pair
(
S
1
,
S
2
)
(S_1,S_2)
(
S
1
,
S
2
)
varies in
F
1
F_1
F
1
. Let the set of all straight line segments joining a pair of distinct points of
E
1
E_1
E
1
and passing through the point R(1, 1) be
F
2
F_2
F
2
.Let
F
2
F_2
F
2
be the set of the mid-points of the line segments in the set
F
2
F_2
F
2
.Then, which of the following statements(s) is (are) TRUE?
[JEE Adv 2018 P2]
The point
(
−
2
,
7
)
(-2,7)
(
−
2
,
7
)
lies in
E
1
E_1
E
1
The point
(
4
5
,
7
5
)
\left(\frac{4}{5},\frac{7}{5}\right)
(
5
4
,
5
7
)
does NOT lie in
E
2
E_2
E
2
The point
(
1
2
,
1
)
\left(\frac{1}{2},1\right)
(
2
1
,
1
)
lies in
E
2
E_2
E
2
The point
(
0
,
3
2
)
\left(0,\frac{3}{2}\right)
(
0
,
2
3
)
does NOT lie in
E
1
E_1
E
1
Validate
Solution:
A
P
=
A
Q
=
A
M
AP=AQ=AM
A
P
=
A
Q
=
A
M
Locus of M is a circle having PQ as its diameter
Hence,
E
1
:
(
x
−
2
)
(
x
+
2
)
+
(
y
−
7
)
(
y
+
5
)
=
0
E_1:(x-2)(x+2)+(y-7)(y+5)=0
E
1
:
(
x
−
2
)
(
x
+
2
)
+
(
y
−
7
)
(
y
+
5
)
=
0
and
x
≠
±
2
x\neq \pm 2
x
=
±
2
Locus of B (midpoint) is a circle having RC as its diameter
E
2
:
x
(
x
−
1
)
+
(
y
−
1
)
2
=
0
E_2:x(x-1)+(y-1)^2=0
E
2
:
x
(
x
−
1
)
+
(
y
−
1
)
2
=
0
Now, after checking the options, we get (D)
© examsnet.com
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