Question

Conjugate surd of $ a - \sqrt b $ is
\[
  A.\,\,\,a + \sqrt b \\
  B.\,\,\sqrt a - b \\
  C.\,\,\sqrt a - \sqrt b \\
  D.\,\,\,\sqrt a + b \\
 \]

Answer 1

Hint: Conjugate surds on product always given a rational number. So, we can use this property to find conjugate surd of given surd. In this we will use option to see which option is correct and hence we can find the correct option or solution of given problem.

Complete step-by-step answer:
Given surd is $ a - \sqrt b $ .
To find its conjugate surd we will check all options given one by one to get the correct option.
Before to get answer to a given problem.
Let's discuss what are conjugate surds?
Conjugate surds are those surds which on product always gives a rational number as a result and they only differ by sign of radical term.
Therefore, to find conjugate surd of given surd we will multiply it with given option one by one to see which option will be a correct option.
Let's start from bottom.
In option (D) we have $ \sqrt a + b $ .
Now, multiplying given surd $ a - \sqrt b $ with option $ \sqrt a + b $ .
We have
 $
\Rightarrow \left( {a - \sqrt b } \right)\left( {\sqrt a + b} \right) \\
   = a\left( {\sqrt a + b} \right) - \sqrt b \left( {\sqrt a + b} \right) \\
   = a\sqrt a + ab - \sqrt {ab} - b\sqrt b \;
  $
Therefore, from above we see that there are too many radical terms in multiplication of given surd $ a - \sqrt b $ with $ \sqrt a + b $ .
Hence, we can say that option (D) will not be the conjugate of given surd.
Therefore option (D) is incorrect.

Now, we will check option (C).
Given term in option (C) is $ \sqrt a - \sqrt b $ .
Multiplying it with given surd we have:
 $
\Rightarrow \left( {a - \sqrt b } \right)\left( {\sqrt a - \sqrt b } \right) \\
   = a\left( {\sqrt a - \sqrt b } \right) - \sqrt b \left( {\sqrt a - \sqrt b } \right) \\
   = a\sqrt a - a\sqrt b - \sqrt {ab} + b \;
  $
Clearly from above we see that $ \sqrt a - \sqrt b $ is not a conjugate surd of given surd $ a - \sqrt b $ .
Therefore, option (C) is the incorrect option.

Now, we will discuss option (B).
In this surd is $ \sqrt a - b $
Multiplying it with given surd we have:
 $
\Rightarrow \left( {a - \sqrt b } \right)\left( {\sqrt a - b} \right) \\
   = a\left( {\sqrt a - b} \right) - \sqrt b \left( {\sqrt a - b} \right) \\
   = a\sqrt a - ab - \sqrt {ab} + b\sqrt b \;
  $
Clearly from above we see that $ \sqrt a - b $ is not a conjugate surd of given surd $ a - \sqrt b $ .
Therefore, option (B) is an incorrect option.

Now, to discuss option (A).
Given surd in option (A) is $ a + \sqrt b $
Multiplying it with given surd we have:
 $
\Rightarrow \left( {a - \sqrt b } \right)\left( {a + \sqrt b } \right) \\
   = a\left( {a + \sqrt b } \right) - \sqrt b \left( {a + \sqrt b } \right) \\
   = {a^2} + a\sqrt b - a\sqrt b - b \\
   = {a^2} - b \;
  $
Hence, the product of two radicals is a rational number. So, we can say that they are conjugate pairs of each other.
Therefore, conjugate pair of given number $ a - \sqrt b \,\,\,is\,\,a + \sqrt b $
Hence, the correct option is option (A).
So, the correct answer is “Option A”.

Note: We can also find conjugate of given surd without any calculation as we know that conjugate surd has the same term but just differs by radical term sign. Therefore, we can say that if a given surd is $ m + \sqrt n $ then its conjugate surd will be given as $ m - \sqrt n $ . But this we can only use for objective problems.