Q.22Δ OAB is formed by lines 2 - 4xy + 2 = 0 and the line x + y - 2 = 0. Find the equation of the median of the triangle drawn from O.
Answers
Answer:
Let D be the midpoint of seg AB where A is (x
1
,y
1
) and B is (x
2
,y
2
)
Then D has coordinates (
2
x
1
+x
2
,
2
y
1
+y
2
)
The joint (combined) equation of the lines OA and OB is x
2
−4xy+y
2
=0 and the equation of the line AB is 2x+3y−1=0
∴ points A and B satisfy the equation 2x+3y−1=0 and x
2
−4xy+y
2
=0 simultaneously.
We eliminate x from the above equations, i.e. put x=
2
1−3y
in the equation x
2
−4xy+y
2
=0, we get,
∴(
2
1−3y
)
2
−4(
2
1−3y
)y+y
2
=0
∴(1−3y)
2
−8(1−3y)y+4y
2
=0
∴1−6y+9y
2
−8y+24y
2
+4y
2
=0
∴37y
2
−14y+1=0
The roots y
1
and y
2
of the above quadratic equation are the y− coordinates of the points A and B.
∴y
1
+y
2
=−
a
b
=
37
14
∴y− coordinates of D=
2
y
1
+y
2
=
37
7
Since D lies on the line AB, we can find the x− coordinate of D as
2x+3(
37
7
)−1=0
∴2x=1−
37
21
=
37
16
∴x=
37
8
∴D is (
37
8
,
37
7
)
∴ equation of the median OD is
8/37
x
=
3/37
y
i.e 7x−8y=0
hope it's helpful for you to learn
make me as brain list