Find the equation of the circumcircle of the triangle formed by the straight line given in the following 5x-3y+4=0
, 2x+3y-5=0, x+y=0.(inter 2nd year maths 2b)
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Answer:
MATHS
Find the equation of the circum circle of the triangle formed by the lines
5x−3y+4=0, 2x+3y−5=0, x+y=0
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ANSWER
Let AB⇒5x−3y+4=0 ......... (1)
BC⇒2x+3y−5=0 ......... (2)
CA⇒x+y=0 ........... (3)
5x−3y=−4
−5x+5y=0
____________
−8y=−4⇒y=
2
1
∴
A(−
2
1
,
2
1
)
2
1
+x=0⇒
x=−
2
1
2x+3y=5
2x+2y=0
____________
y=5
a=−5
∴C(−5,5)
5x−3y=−4
2x+3y=5
___________
7x=1⇒
x=
7
1
∴
(
7
1
,
7
11
)
5(
7
1
)−3y=−4⇒
y=
7
11
Equation of circles is x
2
+y
2
+2gx+2fy+c=0 .......... (4)
Putting A(−
2
1
,
2
1
),B(
7
1
,
7
11
) and c(−5,5) conrequently.
4
1
+
4
1
−g+f+c=0⇒
−g+f+c=−
2
1
..... (5)
49
1
+
49
121
+
7
2
g+
7
22
f+c=0⇒
14g+154f+7c=−122
.......... (6)
25+25−10g+10f+c=0⇒
−10g+10f+c=−50
....... (7)
10gg+10f+10c=−5
−10g+10f+c=−50
_________________
9x=45⇒
c=5
−g+f=−
2
1
−5=−
2
7
⇒
−2g+2f=−7
14g+154f+35=−122⇒
14g+154f=−157
−14g+14f=−49
14g+154f=−157
_______________
168f=−206⇒
f=
84
103
g=
84
397
∴ Equation of circle ⇒84x
2
+84y
2
+794x+206y+420=0
or
42x
2
+42y
2
+397c+103y+210=0