Q60 / 120 · VITEEE 2007 · Mathematics

The differential equation of the system of all circles of radius $r$ in the $X Y$ plane is

  • A
    $\left[1+\left(\frac{\mathrm{d} y}{\mathrm{dx}}\right)^{3}\right]^{2}=r^{2}\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2}$
  • B
    $\left[1+\left(\frac{\mathrm{d} y}{\mathrm{dx}}\right)^{3}\right]^{2}=r^{2}\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{3}$
  • Correct answer
    C
    $\left[1+\left(\frac{\mathrm{d} y}{\mathrm{dx}}\right)^{2}\right]^{3}=r^{2}\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{d} \mathrm{x}^{2}}\right)^{2}$
  • D
    $\left[1+\left(\frac{d y}{d x}\right)^{2}\right]^{3}=r^{2}\left(\frac{d^{2} y}{d x^{2}}\right)^{3}$

Solution

The equation of the family of circles of radius $r$ is $(x-a)^{2}+(y-b)^{2}=r^{2}$ ...(1) Where a \& b are arbitrary constants. Since equation (1) contains two arbitrary constants, we differentiate it two times w.r.t $x \&$ the differential equation will be of second order. Differentiating (1) w.r.t. $x$, we get $\begin{array}{l} 2(x-a)+2(y-b) \frac{d y}{d x}=0 \\ \Rightarrow(x-a)+(y-b) \frac{d y}{d x}=0 ...(2) \end{array}$ Differentiating $(2)$ w.r.t. $x$, we get $\begin{array}{l} 1+(y-b) \frac{d^{2} y}{d x^{2}}+\left(\frac{d y}{d x}\right)^{2}=0 ...(3) \\\ \Rightarrow(y-b)=-\frac{1+\left(\frac{d y}{d x}\right)^{2}}{\frac{d^{2} y}{d x^{2}}} ...(4) \end{array}$ On putting the value of $(y-b)$ in equation (2), we get $x-a=\frac{\left[1+\left(\frac{d y}{d x}\right)^{2}\right] \frac{d y}{d x}}{\frac{d^{2} y}{d x^{2}}}$ ...(5) Substituting the values of $(x-a) \&(x-b)$ in (1), we get $\begin{array}{l} {\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{2}\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}} \\ \left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2}+\frac{\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{2}}{\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2}}=\mathrm{r}^{2} \\ \Rightarrow\left[1+\left(\frac{\mathrm{dy}}{\mathrm{dx}}\right)^{2}\right]^{3}=\mathrm{r}^{2}\left(\frac{\mathrm{d}^{2} \mathrm{y}}{\mathrm{dx}^{2}}\right)^{2} \end{array}$

Hence, the correct answer is option: 'C'.

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