Form the differential equation representing the family of curves given by:
\((x-α)^2+2y^2=α^2\) where a is an arbitrary constant.
Form the differential equation representing the family of curves given by:
\((x-α)^2+2y^2=α^2\) where a is an arbitrary constant.
Solution and Explanation
\((x-α)^2+2y^2=α^2\)
\(⇒x^2+α^2-2αx+2y^2=α^2\)
\(⇒2y^2=2αx-x^2\) ...(1)
Differentiating with respect to \(x\), we get:
\(2y\frac {dy}{dx} =\frac {2α-2x}{2}\)
\(⇒\frac {dy}{dx}=\frac {α-x}{2y}\)
\(⇒\frac {dy}{dx} = \frac {2αx-2x^2}{4xy} \) ...(2)
From equation(1), we get:
\(2αx=2y^2+x^2\)
On substituting this value in equation (3), we get:
\(\frac {dy}{dx}=\frac {2y^2+x^2-2x^2}{4xy}\)
\(⇒\frac {dy}{dx} =\frac {2y^2-x^2}{4xy}\)
Hence, the differential equation of the family of curves is given as \(\frac {dy}{dx} =\frac {2y^2-x^2}{4xy}\).
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