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Byju's Answer
Standard XII
Mathematics
Properties of Argument
If the three ...
Question
If the three points
(
a
,
b
)
,
(
a
+
r
cos
α
,
b
+
r
sin
α
)
,
(
a
+
r
cos
β
,
b
+
r
sin
β
)
are the vertices of an equilateral triangle. Find the value of
|
α
−
β
|
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Solution
Let
A
(
a
,
b
)
;
B
(
a
+
r
cos
α
,
b
+
sin
α
)
;
C
(
a
+
r
cos
β
,
b
+
r
sin
β
)
We have
⇒
A
C
=
√
(
a
+
r
cos
β
−
a
)
2
+
(
b
+
r
sin
β
−
b
)
2
⇒
A
C
=
√
(
r
2
cos
2
β
)
+
(
r
2
sin
2
β
)
=
√
r
2
=
r
⇒
A
C
=
r
Now,
⇒
B
C
=
√
(
a
+
r
cos
α
−
a
−
r
cos
β
)
2
+
(
b
+
r
sin
α
−
b
−
r
sin
β
)
2
⇒
B
C
=
√
r
2
(
cos
α
−
cos
β
)
2
+
r
2
(
sin
α
−
sin
β
)
2
⇒
B
C
=
r
√
(
cos
2
α
+
cos
2
β
−
2
cos
α
cos
β
)
+
(
sin
2
α
+
sin
2
β
−
2
sin
α
sin
β
)
⇒
B
C
=
r
√
1
+
1
−
2
cos
α
cos
β
−
2
sin
α
sin
β
⇒
B
C
=
r
√
2
−
2
(
cos
α
cos
β
+
sin
α
sin
β
)
⇒
B
C
=
r
√
2
−
2
cos
(
α
−
β
)
As
△
A
B
C
is equilateral triangle,
A
C
=
B
C
⇒
r
√
2
−
2
cos
(
α
−
β
)
=
r
⇒
√
2
−
2
cos
(
α
−
β
)
=
1
⇒
2
−
2
cos
(
α
−
β
)
=
1
⇒
cos
(
α
−
β
)
=
1
2
Therefore,
|
α
−
β
|
=
π
3
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Similar questions
Q.
Find whether the points (b, b), (−b, −b) and (2b, 2b) are the vertices of an equilateral triangle.
Q.
Show that the points (a, b, c), (b, c, a) and (c, a, b) are the vertices of an equilateral triangle.
Q.
Assertion :Points
P
(
−
sin
(
β
−
α
)
,
−
cos
β
)
,
Q
(
cos
(
β
−
α
)
,
sin
β
)
and
R
(
cos
(
β
−
α
+
θ
)
,
sin
(
β
−
θ
)
)
, where
β
=
π
4
+
α
2
are non-collinear. Reason: Three given points are non-collinear if they form a triangle of non-zero area.
Q.
Suppose that
z
1
,
z
2
,
z
3
are three vertices of an equilateral triangle in the argand plane. Let
α
=
1
2
(
√
3
+
i
)
and
β
be an non-zero complex number. The points
α
z
1
+
β
,
α
z
2
+
β
,
α
z
3
+
β
will be
Q.
If the points
A
(
z
)
,
B
(
−
z
)
and
C
(
1
−
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)
are the vertices of an equilateral triangle
A
B
C
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