Question

Form the differential equation of the family of curves represented by the equation (a being the parameter):
(2x − a)² − y² = a²

Answer 1

The equation of the family of curves is \[\left( 2x - a \right)^2 - y^2 = a^2 \]
\[ \Rightarrow 4 x^2 - 4ax + a^2 - y^2 = a^2 \]
\[ \Rightarrow 4 x^2 - 4ax - y^2 = 0 . . . \left( 1 \right)\]
where a is a parameter.
As this equation has only one parameter, we shall get a differential equation of first order.
Differentiating (1) with respect to x, we get
\[8x - 4a - 2y\frac{dy}{dx} = 0\]
\[ \Rightarrow - y\frac{dy}{dx} + 4x = 2a . . . \left( 2 \right)\]
Now, from (1), we get
\[2a = \frac{4 x^2 - y^2}{2x}\]                                     ...(3)
From (2) and (3), we get
\[- y\frac{dy}{dx} + 4x = \frac{4 x^2 - y^2}{2x}\]
\[ \Rightarrow - 2xy\frac{dy}{dx} + 8 x^2 = 4 x^2 - y^2 \]
\[ \Rightarrow - 2xy\frac{dy}{dx} + 4 x^2 + y^2 = 0\]
\[ \Rightarrow 2xy\frac{dy}{dx} = 4 x^2 + y^2 \]
It is the required differential equation.