the point where straight line x/a+y/b =1 meets x-axis is
a)(0,a)
b) (a,0)
c) (0,b)
d) ( b,0)
Answers
Step-by-step explanation:
ANSWER
we have,
a
x
+
b
y
=1and
b
x
+
a
y
=1
⇒bx+ay−ab=0⟶(1)
and,ax+by−ab=0⟶(2)
Equation of line through their point of intersection will be given by:
(bx+ay−ab)+k(ax+by−ab)=0
where k is constant
⇒x(b+ak)+y(a+bk)−ab(k−1)=0
Line meet x-axis at A, hence for A, y=0
⇒x(b+ak)−ab(k−1)=0
⇒x=
(b+ak)
ab(k−1)
so, coordinate of A is (
(b+ak)
ab(k−1)
,0)
Line meets y-axis at B, hence for B, x=0
⇒y(a+bk)−ab(k−1)=0
⇒y=
(a+bk)
ab(k−1)
so, coordinate of B is (0,
(a+bk)
ab(k−1)
)
Let (h,m) be midpoint of AB, hence
h=
2
(b+ak)
ab(k−1)
+0
⇒k=
a
b
(
b−2h
2h+a
)⟶(3)
and,m=
2
0+
a+bk
ab(k−1)
⇒k=
b
a
(
a−2m
2m+b
)⟶(4)
From (3) and (4) we get
⇒
a
b
(
b−2h
2h+a
)=
b
a
(
a−2m
2m+b
)
To get equation of locus we take h→xandm→y
⇒
a
b
(
b−2x
2x+a
)=
b
a
(
a−2y
2y+b
)
⇒2xy(a+b)=ab(x+y)