Question

Solve the following equation.
\[\dfrac{\mathrm{x+}\sqrt{{{\mathrm{x}}^{\mathrm{2}}}\mathrm{-1}}}{\mathrm{x-}\sqrt{{{\mathrm{x}}^{\mathrm{2}}}\mathrm{-1}}}\mathrm{+}\dfrac{\mathrm{x-}\sqrt{{{\mathrm{x}}^{\mathrm{2}}}\mathrm{-1}}}{\mathrm{x+}\sqrt{{{\mathrm{x}}^{\mathrm{2}}}\mathrm{-1}}}\mathrm{=98}\]
A. ±1
B. ±5
C. ±10
D. ±4

Answer 1

Hint: First of all we will rationalize the given fraction by multiplying its numerator as well as denominator by the conjugate of its denominator, i.e., the algebraic sign changed between rational and irrational part.

Complete step-by-step solution:
In this type of question, first of all, we have to rationalize the denominator part of the fraction so that we can easily perform the mathematical operation between them.
Now, doing rationalization of fraction having irrational part in denominator is as shown below;
\[\dfrac{a~+\sqrt{b}}{a-\sqrt{b}}\times \dfrac{a+\sqrt{b}}{a+\sqrt{b}}=\dfrac{{{a}^{2}}+b+2a\sqrt{b}}{{{a}^{2}}-b}\]
Here, we can see that the denominator as well as the numerator is multiplied by the conjugate of the denominator, i.e., the algebraic sign changed between the rational and irrational part.
Similarly, we will rationalize the given expression;
\[\Rightarrow \dfrac{x+\sqrt{{{x}^{2}}-1}}{x-\sqrt{{{x}^{2}}-1}}\times \dfrac{x+\sqrt{{{x}^{2}}-1}}{x+\sqrt{{{x}^{2}}-1}}+\dfrac{x-\sqrt{{{x}^{2}}-1}}{x+\sqrt{{{x}^{2}}-1}}\times \dfrac{x-\sqrt{{{x}^{2}}-1}}{x-\sqrt{{{x}^{2}}-1}}=98\]
Now, we will further solve the above equation and we get ;
\[\Rightarrow \dfrac{{{x}^{2}}+{{x}^{2}}-1+2x\sqrt{{{x}^{2}}-1}}{{{x}^{2}}-{{x}^{2}}+1}+\dfrac{{{x}^{2}}+{{x}^{2}}-1-2x\sqrt{{{x}^{2}}-1}}{{{x}^{2}}-{{x}^{2}}+1}=98\]
Again, we will simplify the numerator and denominator, we get ;
\[\Rightarrow \dfrac{2{{x}^{2}}-1+2x\sqrt{{{x}^{2}}-1}}{1}+\dfrac{2{{x}^{2}}-1-2x\sqrt{{{x}^{2}}-1}}{1}=98\]
After adding and subtracting the same degree term of the above expression, we get ;
\[\Rightarrow \dfrac{4{{x}^{2}}-2}{1}=98\]
Now, we will simplify the above expression;
\[\begin{align}
  & \Rightarrow 4{{x}^{2}}=98+2 \\
 & \Rightarrow {{x}^{2}}=\dfrac{100}{4} \\
 & \Rightarrow {{x}^{2}}=25 \\
 & \Rightarrow x=\pm 5 \\
\end{align}\]
Therefore, the correct answer of the above question will be option (B).

Note: Always remember the point of rationalization of a fraction as shown above in which denominator having irrational part will be helpful in all these kinds of questions.
Also, be careful while doing rationalization because there is a chance of calculation error in multiplication by the conjugate. Since the options are given so on the basis of a competitive examination view, we can also find the correct option by putting the value in the expression in less time.