Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
(x − a)² − y² = 1

Answer 1

The equation of family of curves is \[\left( x - a \right)^2 - y^2 = 1.........(1)\]

where a is a parameter.

As this equation has only one arbitrary constant, we shall get a differential equation of first order.

Differentiating (1) with respect to x, we get

\[2\left( x - a \right) - 2y\frac{dy}{dx} = 0\]

\[ \Rightarrow \left( x - a \right) - y\frac{dy}{dx} = 0\]

\[ \Rightarrow \sqrt{1 + y^2} = y\frac{dy}{dx} ........\left[\text{Using}\left( 1 \right) \right]\]

\[ \Rightarrow 1 + y^2 = y^2 \left( \frac{dy}{dx} \right)^2 \]

\[ \Rightarrow y^2 \left( \frac{dy}{dx} \right)^2 - y^2 = 1\]

It is the required differential equation.