Question

If x, y and z are in A.P. then \[\dfrac{1}{{\sqrt x + \sqrt y }},\dfrac{1}{{\sqrt z + \sqrt x }},\dfrac{1}{{\sqrt y + \sqrt z }}\] are in
A. A.P.
B. G.P.
C. H.P.
D. A.P. and H.P.

Answer 1

Hint: The problem is looking very troubling but it is very easy to solve. As we are given that x, y and z are in A.P. we will use this data only with the help of which we can reach the answer. Also we should know what is A.P., G.P. and H.P. Now let’s start solving.

Complete step by step answer:
Given that, x, y and z are in A.P.
We can write,
\[y - x = z - y\]
Multiplying both sides by a minus sign,
\[x - y = y - z\]
Taking the reciprocal,
\[\dfrac{1}{{x - y}} = \dfrac{1}{{y - z}}\]
Now this can be written as,
\[\dfrac{1}{{{{\left( {\sqrt x } \right)}^2} - {{\left( {\sqrt y } \right)}^2}}} = \dfrac{1}{{{{\left( {\sqrt y } \right)}^2} - {{\left( {\sqrt z } \right)}^2}}}\]
On expanding the brackets,
\[\dfrac{1}{{\left( {\sqrt x - \sqrt y } \right)\left( {\sqrt x + \sqrt y } \right)}} = \dfrac{1}{{\left( {\sqrt y - \sqrt z } \right)\left( {\sqrt y + \sqrt z } \right)}}\]
Now take the negative terms on numerator,
\[\dfrac{{\sqrt y - \sqrt z }}{{\sqrt x + \sqrt y }} = \dfrac{{\sqrt x - \sqrt y }}{{\sqrt y + \sqrt z }}\]
Now we need \[\sqrt z + \sqrt x \] in the denominator on LHS and RHS both. For that we will add and subtract \[\sqrt x \] on LHS and \[\sqrt z \] on RHS.
\[\dfrac{{\sqrt x + \sqrt y - \sqrt z - \sqrt x }}{{\sqrt x + \sqrt y }} = \dfrac{{\sqrt z + \sqrt x - \sqrt y - \sqrt z }}{{\sqrt y + \sqrt z }}\]
Now taking the minus sign common from last two terms,
\[\dfrac{{\sqrt x + \sqrt y - \left( {\sqrt z + \sqrt x } \right)}}{{\sqrt x + \sqrt y }} = \dfrac{{\sqrt z + \sqrt x - \left( {\sqrt y + \sqrt z } \right)}}{{\sqrt y + \sqrt z }}\]
On dividing both the sides by \[\sqrt z + \sqrt x \] and then separating the terms we get,
\[\dfrac{1}{{\sqrt z + \sqrt x }} - \dfrac{1}{{\sqrt x + \sqrt y }} = \dfrac{1}{{\sqrt y + \sqrt z }} - \dfrac{1}{{\sqrt z + \sqrt x }}\]
Now we can easily say that the terms are in A.P. because the difference between first and second (we can say) is equal to difference between third and second.
thus \[\dfrac{1}{{\sqrt x + \sqrt y }},\dfrac{1}{{\sqrt z + \sqrt x }},\dfrac{1}{{\sqrt y + \sqrt z }}\] are in A.P.

So, the correct answer is “Option A”.

Note: Note that given terms were in A.P. need not necessarily mean that the asked terms should be in A.P. that is just a coincidence. Sometimes they can be in G.P. that is they can have a common ratio instead of common difference. Also they can be in H.P. that is the middle term is the harmonic mean of the neighboring terms.
This totally depends on the format of the terms so asked.