Question

If (a+b) varies as (a-b), prove that a2+b2 varies as b.

Answer 1

a2+b2anda2−b2
a
2
+
b
2
and
a
2
-
b
2
both (a) and (b)
None of these
Answer :
C
Solution :
Given that a + b vary directly with a - b.
a+ba−b=k
a
+
b
a
-
b
=
k
(constant)
Applying componendo - dividendo theorem,
ab=k+1k−1
a
b
=
k
+
1
k
-
1
as k is constant, k+1k−1
k
+
1
k
-
1
is also a constant.
∴
∴
a and b vary directly.
Let k+1k−1
k
+
1
k
-
1
be p
⇒ab=p⇒a2b2=p2
⇒
a
b
=
p
⇒
a
2
b
2
=
p
2
.
By applying componendo-dividendo theorem,
a2+b2a2−b2=p2+1p2−1
a
2
+
b
2
a
2
-
b
2
=
p
2
+
1
p
2
-
1
.
As p is a constant, p2+1p2−1
p
2
+
1
p
2
-
1
is also a consonant.
∴a2+b2anda2−b2
∴
a
2
+
b
2
and
a
2
-
b
2
vary directly.