Question

\(\underline{\text { Assertion - Reason Type Questions: }}\) Each question has four choices (a), (b), (c) and (d) out of which only one is correct. Write (a), (b), (c) and (d) according to the following rules. (a) Statement- 1 is True, Statement-2 is True, Statement- 2 is a correct explanation for Statement-1. (b) Statement-1 is True, Statement- 2 is True, Statement-2 is not a correct explanation for Statement-1. (c) Statement- 1 is True, Statement- 2 is False. (d) Statement- 1 is False, Statement- 2 is True. Statement \(-2:\) The differential equation \(\mathrm{y}^{\prime}=(\mathrm{y} / 2 \mathrm{x})\) is variable separable. Statement-1: Curve satisfying the differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) passing through \((2,1)\) is a parabola with Focus \([(1 / 4), \underline{0}]\). Statement- 2 : The differential equation \((\mathrm{dy} / \mathrm{dx})=(\mathrm{y} / 2 \mathrm{x})\) is variable separable.

Short answer

(d) Statement-1 is False, Statement-2 is True.

Step-by-step solution

Verify Statement-2

Determine if the given differential equation is indeed variable separable. The given differential equation is: \[\frac{dy}{dx} = \frac{y}{2x}\] To check if it is variable separable, we can try to rewrite the equation as a product of functions of x and y: \[\frac{dy}{y} = \frac{dx}{2x}\] Since we can do this, we can confirm that statement-2 is true and the given differential equation is variable separable.

Solve the differential equation

Now that we know the equation is variable separable, we can proceed with solving the differential equation by integrating both sides: \[\int{\frac{dy}{y}} = \int{\frac{dx}{2x}}\] This gives: \[\ln{y} = \frac{1}{2}\ln{x} + C\] To find the constant C, we use the given point (2,1) through which the curve passes: \[\ln{1} = \frac{1}{2}\ln{2} + C\] Since \(\ln{1}=0\), we have \(C = -\frac{1}{2}\ln{2}\). Now, we can rewrite the equation as: \[y = x^{\frac{1}{2}}e^{-\frac{1}{2}\ln{2}}\] By applying the properties of logarithms and exponentials, we can simplify this as: \[y = \sqrt{\frac{x}{2}}\]

Verify Statement-1

Now that we have the curve equation, let's check if it is a parabola with the given focus: \[y = \sqrt{\frac{x}{2}}\] To represent a parabola with focus at \(\left(\frac{1}{4}, 0\right)\), the equation must satisfy the definition of a parabola: the distance from a point on the curve to the focus is equal to its distance to the directrix. In the case of a parabola with the given focus, the directrix will be a vertical line located to the left of the focus. The curve we found does not follow this parabola definition, as it is not symmetric and does not have the same distances to the focus and directrix. Therefore, statement-1 is false. To summarize: - Statement-1 is False - Statement-2 is True Thus, the correct answer for this exercise is: (d) Statement-1 is False, Statement-2 is True.

Key concepts

Differential Equations

Differential equations are mathematical equations that involve derivatives of a function with respect to one or more variables. These derivatives represent rates of change, making differential equations powerful tools for modeling real-world phenomena where change is a constant.
To understand differential equations, it is crucial to recognize the different types. There are ordinary differential equations (ODEs), which involve functions of a single variable and their derivatives, and partial differential equations (PDEs), which involve multiple variables.
In the context of ordinary differential equations, like the one given in the exercise \[\frac{dy}{dx} = \frac{y}{2x},\]we deal with relationships involving only one independent variable, usually time or space. Successfully solving these equations typically involves finding a function or a set of functions that satisfies the equation, known as the solution of the equation.

Variable Separable Method

The variable separable method is a practical technique for solving a class of differential equations. These are equations that can be expressed as a product of two separate functions, each depending on one of the variables.
The key to using this method is to rewrite the given differential equation in the form\[\frac{dy}{g(y)} = \frac{dx}{h(x)},\]where one side is solely dependent on the variable \( y \) and the other solely on \( x \). This separation allows us to integrate both sides independently.

• Identify if the equation is separable by rearranging terms.
• Once separable, integrate both sides to find a general solution.
• Apply initial conditions, if provided, to determine any constants of integration.

In the problem statement, the differential equation \( \frac{dy}{dx} = \frac{y}{2x} \) was successfully separated and integrated leading to a solution. This method simplifies the solving process and provides an analytical solution to the equation.

Parabola

A parabola is a symmetric, U-shaped curve formed by all points that are equidistant from a point called the focus and a line called the directrix. Parabolas are a specific type of conic section and are defined mathematically by equations of the form\[ y = ax^2 + bx + c, \]where \( a, b, \) and \( c \) are constants.
In the context of the exercise, there was a claim that the curve described by the differential equation results in a parabola with a specific focus. Upon verification, the equation\[ y = \sqrt{\frac{x}{2}} \]did not satisfy the conditions of a parabola such as symmetry and the specific geometric property relating to the focus and the directrix.

• The focus is a fixed point about which the distances of the points on the parabola are calculated.
• A true parabola has points equidistant to its focus and its directrix.
• This strict definition was not met in the solution provided in the exercise, ruling out the possibility of the curve being a true parabola with the defined focus.