Question

Represent the following families of curves by forming the corresponding differential equations (a, b being parameters):
x² + (y − b)² = 1

Answer 1

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

where `b` is a parameter.

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

Differentiating (1) with respect to x, we get

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

\[ \Rightarrow 2x + 2\sqrt{1 - x^2}\frac{dy}{dx} = 0 .......\left[ \text{Using }\left( 1 \right) \right]\]

\[ \Rightarrow x = - \sqrt{1 - x^2}\frac{dy}{dx}\]

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

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

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

It is the required differential equation.