Question

The differential equation of family of hyperbolas with asymptotes \(\mathrm{x}+\mathrm{y}=1\) and \(\mathrm{x}-\mathrm{y}=1\) is: (A) \(\mathrm{yy}_{1}=\mathrm{x}-1\) (B) \(\mathrm{yy}_{1}+\mathrm{x}=0\) (C) \(\mathrm{yy}_{2}=\mathrm{y}_{1}\) (D) \(\mathrm{y}_{1}+\mathrm{xy}=0\)

Short answer

The correct answer is (A). The differential equation of the family of hyperbolas with asymptotes \(\mathrm{x}+\mathrm{y}=1\) and \(\mathrm{x}-\mathrm{y}=1\) is \(\mathrm{yy}_{1}=\mathrm{x}-1\).

Step-by-step solution

Convert the asymptotes equations to a usable form for the hyperbola equation

We have the asymptotes given by \(\mathrm{x}+\mathrm{y}=1\) and \(\mathrm{x}-\mathrm{y}=1\). We need to convert these equations into the form of the equation of the hyperbola. First, add the two equations to eliminate \(\mathrm{y}\), and then subtract the two equations to eliminate \(\mathrm{x}\): \[\mathrm{x}+\mathrm{y}=1\implies \mathrm{y} = 1-\mathrm{x}\] \[\mathrm{x}-\mathrm{y}=1\implies \mathrm{y} = \mathrm{x}-1\] Now, let's compute the square of each equation and then add both to eliminate \(\mathrm{y}^2\): \[\left(1-\mathrm{x}\right)^2 =1-2\mathrm{x}+\mathrm{x}^2\] \[\left(\mathrm{x}-1\right)^2 =\mathrm{x}^2-2\mathrm{x}+1\]

Find the equation of the hyperbola

Adding the two squares of the equations obtained in step 1, we get: \[1-2\mathrm{x}+\mathrm{x}^2+\mathrm{x}^2-2\mathrm{x}+1=\mathrm{y}^2\] \[2\mathrm{x}^2-4\mathrm{x}+2=\mathrm{y}^2\] \[2(\mathrm{x}^2-2\mathrm{x}+1)=\mathrm{y}^2\] The resulting equation represents the equation of the family of hyperbolas.

Differentiate the hyperbola equation

Now, we need to differentiate the equation \(2(\mathrm{x}^2-2\mathrm{x}+1)=\mathrm{y}^2\) with respect to \(\mathrm{x}\) to obtain the differential equation representing the family of hyperbolas: \[2(2\mathrm{x}-2)\frac{d\mathrm{x}}{d\mathrm{x}}=2\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}}\] \[4(\mathrm{x}-1)=\mathrm{y}\frac{d\mathrm{y}}{d\mathrm{x}}\] \[\mathrm{y}_{1}=\frac{4(\mathrm{x}-1)}{\mathrm{y}}\]

Choose the correct answer

Comparing it with the given options, we can see that the obtained differential equation matches the option (A): \[\mathrm{yy}_{1}=\mathrm{x}-1\] Therefore, the correct answer is (A).

Key concepts

Asymptote of Hyperbola

Understanding the concept of an asymptote is crucial when studying hyperbolas. An \textbf{asymptote} is essentially a line that a hyperbola approaches as it extends into infinity but never actually reaches. These lines provide a frame of reference and define the ultimate shape of the hyperbola.

In the given exercise, the hyperbola is described as having two asymptotes: \(\mathrm{x} + \mathrm{y} = 1\) and \(\mathrm{x} - \mathrm{y} = 1\). These lines do not intersect the hyperbola; instead, they represent the direction in which the arms of the hyperbola extend as they grow infinitely. This concept is important as it also provides part of the structure from which we can derive the equation of the family of hyperbolas.

When creating graphical representations of hyperbolas, sketching in the asymptotes provides a guide for drawing the curve. The arms of the hyperbola will approach these lines more closely the further they extend from the center.

Family of Hyperbolas

The term \textbf{family of hyperbolas} refers to a set of hyperbolas that share common asymptotes or other features but differ in size, shape, or position. The exercise deals specifically with a family of hyperbolas defined by their common asymptotes. What distinguishes members of a family from one another are the constants present in the standard hyperbola equation \(\frac{x^2}{a^2} - \frac{y^2}{b^2} = 1\), where the variables \(a\) and \(b\) affect the 'spread' of the hyperbola.

To determine the differential equation for a family of hyperbolas sharing asymptotes, as in the given problem, it is necessary to manipulate and differentiate the equations of the asymptotes. By doing so, we can find a general equation that represents all possible hyperbolas within the family, allowing various solutions based on the initial conditions or specific features.

Differentiation for Hyperbola Equations

Differentiation is a mathematical process used to find the rate at which one variable changes with respect to another. When applied to the equation of a hyperbola, \textbf{differentiation} yields a differential equation that describes the slope of the tangent to any point on the hyperbola.

In this exercise, after finding the equation representing the family of hyperbolas, \(2(\mathrm{x}^2 - 2\mathrm{x} + 1) = \mathrm{y}^2\), we differentiate it with respect to \(\mathrm{x}\) to get the differential equation. This step is vital to connecting a point on the curve with its slope, and hence is useful for understanding the behavior of the hyperbolas across different intervals. The differentiation helps in establishing a relationship between \(\mathrm{x}\), \(\mathrm{y}\), and the derivative \(\mathrm{y}_1 = \frac{d\mathrm{y}}{d\mathrm{x}}\), which is necessary to deduce the correct differential equation of the hyperbola family.