if $e^x \, =\, y+ \sqrt{1+y^2},$ then the value of y is
- $\frac{1}{2}(e^x+e^{-x})$
- $\frac{1}{2}(e^x - e^{-x})$
- $e^x \, - \, e ^{\frac{-x}{2}}$
- $e^x \, + \, e ^{\frac{-x}{2}}$
The Correct Option is B
Solution and Explanation
Given \(e^x =\, y + \sqrt{1+y^2}\)
\(\Rightarrow \, \, e^x \, - y = \sqrt{1+y^2}\)
Squaring both side, we have
\(e^{2x} + y^2 - 2e^xy \, = 1+y^2\)
\(\Rightarrow 2e^{x}y=e^{2x}-1\)
\(\Rightarrow \, \, y=\frac{e^{2x}-1}{2e^x}\)
\(\Rightarrow \, y=\frac{1}{2} \, [e^x - e^{-x}]\)
So, the correct answer is (B): \(\Rightarrow \, y=\frac{1}{2} \, [e^x - e^{-x}]\)
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Concepts Used:
Solution of System of Linear Inequalities in Two Variables
A System of Linear Inequalities is a set of 2 or more linear inequalities which have the same variables.
Example
\(x + y ≥ 5\)
\(x – y ≤ 3\)
Here are two inequalities having two same variables that are, x and y.
Solution of System of Linear Inequalities in Two Variables
The solution of a system of a linear inequality is the ordered pair which is the solution of all inequalities in the studied system and the graph of the system of a linear inequality is the graph of the common solution of the system.
Therefore, the Solution of the System of Linear Inequalities could be:
Graphical Method:
For the Solution of the System of Linear Inequalities, the Graphical Method is the easiest method. In this method, the process of making a graph is entirely similar to the graph of linear inequalities in two variables.
Non-Graphical Method:
In the Non-Graphical Method, there is no need to make a graph but we can find the solution to the system of inequalities by finding the interval at which the system persuades all the inequalities.
In this method, we have to find the point of intersection of the two inequalities by resolving them. It could be feasible that there is no intersection point between them.