If x-y=2 is the equation of a chord of the circle x^2 + y^2 +2y=0. Find the equation of the circle of which this chord is a diameter.
Answers
Answer:
Given that
the equation of the chord
x
−
y
=
2
...
.
.
[
1
]
the equation of the circle
x
2
+
y
2
+
2
y
=
0
...
...
[
2
]
From these two equations we get
(
y
+
2
)
2
+
y
2
+
2
y
=
0
⇒
2
y
2
+
6
y
+
4
=
0
⇒
2
y
2
+
6
y
+
4
=
0
⇒
y
2
+
3
y
+
2
=
0
⇒
y
2
+
2
y
+
y
+
2
=
0
⇒
y
(
y
+
2
)
+
1
(
y
+
2
)
=
0
⇒
(
y
+
2
)
(
y
+
1
)
=
0
So
y
=
−
1
and
−
2
Inserting in [1] we get
x
=
1
and
0
respectively
So the coordinates of points intersections of the chord with the circles are
(
1
,
−
1
)
and
(
0
,
−
2
)
The line segment of the chord is also the diameter of a circle.
So equation of this circle having diameter
x
−
y
=
2
will be
y
+
1
x
−
1
×
y
+
2
x
−
0
=
−
1
⇒
(
y
+
1
)
×
(
y
+
2
)
=
−
x
(
x
−
1
)
⇒
x
2
+
y
2
−
x
+
3
y
+
2
=
0
...
.
.
[
3
]
This is the required equation of the circle
Step-by-step explanation: