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Byju's Answer
Standard XII
Mathematics
Chord of Contact
find points o...
Question
find points on the circle x
2
+y
2
=a
2
tangent are drawn to hyperbola x
2
-y
2
=a
2
find locus of middle point of chord of contact
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Solution
The
equation
of
circle
is
:
x
2
+
y
2
=
a
2
.
.
.
1
Any
point
on
the
circle
is
P
a
cos
θ
,
a
sin
θ
.
Equation
of
chord
of
contact
of
the
tangents
from
P
to
hyperbola
x
2
-
y
2
=
a
2
is
x
cos
θ
-
y
sin
θ
=
a
.
.
.
2
Let
M
x
1
,
y
1
be
the
mid
point
of
the
chord
of
contact
2
.
Then
its
equation
is
:
xx
1
-
yy
1
=
x
1
2
-
y
1
2
.
.
.
.
3
Equations
2
and
3
represent
chord
of
contact
.
Comparing
the
coefficients
of
like
terms
in
2
and
3
,
we
get
cos
θ
x
1
=
sin
θ
y
1
=
a
x
1
2
-
y
1
2
⇒
sin
θ
=
ay
1
x
1
2
-
y
1
2
and
cos
θ
=
ax
1
x
1
2
-
y
1
2
Now
,
sin
2
θ
+
cos
2
θ
=
1
⇒
a
2
y
1
2
+
a
2
x
1
2
x
1
2
-
y
1
2
2
=
1
⇒
x
1
2
-
y
1
2
2
=
a
2
x
1
2
+
y
1
2
So
,
locus
of
mid
point
M
is
,
x
2
-
y
2
2
=
a
2
x
2
+
y
2
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0
Similar questions
Q.
From the points on the circle
x
2
+
y
2
=
a
2
, tangents are drawn to the hyperbola
x
2
−
y
2
=
a
2
; the locus of the middle points of the chords of contact is
Q.
From the points of the circle
x
2
+
y
2
=
a
2
, tangents are drawn to the hyperbola
x
2
−
y
2
=
a
2
, then the locus of the middle points of the chords of contact is
Q.
Find the locus of the middle point of the chord of contact of tangents from any point of the circle
x
2
+
y
2
=
r
2
to the hyperbola
x
2
a
2
−
y
2
b
2
=
1
Q.
Tangents are drawn from the points on a tangent of the hyperbola
x
2
−
y
2
=
a
2
to the parabola
y
2
=
4
a
x
. If all the chords of contact pass through a fixed point
Q
, then the locus of the point
Q
for different tangents on the hyperbola is
Q.
Tangents are drawn from any point on the hyperbola
x
2
9
−
y
2
4
=1 to the circle
x
2
+
y
2
=
9
Find the locus of midpoint of the chord of contact
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