show that the following four points in each of the following are concyclic and find the equation of the circle in which they lie
(1,2),(3,-4),(5,-6),(19,8)
Answers
Answer:
Fore concyclic
BD×AC=AD.BC+AB.DC
(1) A(1,1),B(−6,0),C(−2,2),D(−2,−8)
⇒
(−6+2)
2
+(8)
2
.
3
2
+1
2
=
3
2
+9
2
.
4
2
+2
2
+
7
2
+1
.
10
2
+10
2
⇒
16+64
.
10
=
1800
+10
50
⇒20
2
=20
2
Hence they are concyclic
(2) (1,−2),(3,4)(5,−6),(19,8)
⇒
16
2
+12
2
.
4
2
+8
2
=
18
2
+6
2
.
2
2
+6
2
+
2
2
+6
2
14
2
+14
2
⇒
400
.
80
=
360
.
8
+
40
392
⇒80
5
=80
5
Hence there are concyclic.
(3) (1,−6),(5,2),(7,0),(−1,−4)
⇒
6
2
+6
2
.
6
2
+6
2
=
2
2
+2
2
.
2
2
+2
2
+
4
2
+8
2
.
8
2
+4
2
⇒
36+36
.
36+36
=
8
.
8
+
16+64
+
16+64
⇒
72
.
72
=80+8
72=88
Hence they are not concyclic
(4) (9,1),(7,9),(−1,12),(6,10)
⇒
(1)
2
+(1)
2
.
10
2
+11
2
=
3
2
+9
2
.
8
2
+3
2
+
2
2
+8
2
7
2
+20
2
⇒
2
.
221
=
90
73
+
68
53
⇒
442
=
6570
+
3604
⇒
442
=
6629
Hence they are no concyclic