units.Equation of the family of circles having centers on $y$-axis and radius $2$ units can be represented by $(x)^{2}+(y-a)^{2}=4$, where $a$ is an arbitrary constant. $(y-a)^{2}+x^{2}=4(1)$...
at the origin.Equation of the family of all circles touching the $y$-axis at the origin can be represented by $(x-a)^{2}+y^{2}=a^{2}$, where $a$ is an arbitrary constants. $(x-a)^{2}+y^{2}=a^{2}(1)$...
and centers at the origin.Equation of the family of an ellipse having foci on the $y$-axis and centers at the origin can be represented by $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$ $......(1)$ Differentiating the above...
where a is an arbitrary constant.Equation of the family of curves, $x^{2}+y^{2}-2 a y=a^{2}$, where $a$ is an arbitrary constant. $\begin{array}{l} x^{2}-2 a x+a^{2}+2 y^{2}=a^{2} \\ x^{2}-2 a x+2 y^{2}=0 \end{array}$...
and centers at the origin.Equation of the family of an ellipse having foci on the $y$-axis and centers at the origin can be represented by...
where 
is an arbitrary constant.Equation of the family of curves, $(x-a)^{2}+2 y^{2}=a^{2}$, where $a$ is an arbitrary constant. $\begin{array}{l} x^{2}-2 a x+a^{2}+2 y^{2}=a^{2} \\ x^{2}-2 a x+2 y^{2}=0 \end{array}$...
Equation of the family of parabolas having a vertex at the origin and axis along positive $y$-axis can be represented by $(x)^{2}=4 a y$, where $a$ is an arbitrary constants. $x^{2}=4 a y(1)$...
and 
where is the parameter.Now, it is not necessary that the centre of the circle will lie on origin in this case. Hence let us assume the coordinates of the centre of the circle be $(0, \mathrm{~h})$. Here, $\mathrm{h}$ is...
where and 
are parameters.Equation of the family of curves, $y^{2}=m\left(a^{2}-x^{2}\right)$, where $a$ and $m$ are parameters. Differentiating the above equation with respect to $x$ on both sides, we have,...
and radius unity.Equation of the family of circles having centers on the $x$-axis and radius unity can be represented by $(x-a)^{2}+(y)^{2}=1$, where $a$ is an arbitrary constants. $(x-a)^{2}+y^{2}=1(1)$...
where and 
are arbitrary constants.Equation of the family of curves, $y=a e^{b x}$, where $a$ and $b$ are arbitrary constants. Differentiating the above equation with respect to $x$ on both sides, we have, $\begin{array}{l} y=a e^{b...
where 
and 
are arbitrary constants.Equation of the family of curves, $x=A \cos n t+B \sin n t$, where $A$ and $B$ are arbitrary constants. Differentiating the above equation with respect to $t$ on both sides, we have,...
Equation of the family of circles in the second quadrant and touching the coordinate axes can be represented by $(x-(-a))^{2}+(y-a)^{2}=a^{2}$, where $a$ is an arbitrary constants....
Where and 
are parameters.Equation of the family of curves, $y=a \sin (b x+c)$, Where $a$ and $c$ are parameters. Differentiating the above equation with respect to $x$ on both sides, we have, $\begin{array}{l} y=a \sin (b...
X^2+y^2=a^2 Differentiating w.r.t x, 2x+2ydx/dy=0. Now as a is eliminated so this is the differential equation of the family of concentric circles. ⇒dx/dy=−y/x
where and are arbitrary constants.The equation of a straight line is represented as, $Y=m x+c$ Differentiating the above equation with respect to $x$, $\frac{\mathrm{dy}}{\mathrm{dx}}=\mathrm{m}$ Differentiating the above equation...