Question

The differential equation whose solution is $(x-h{)}^2+(y-k{)}^2=a^2$ ( $a$ is a constant), is

• A. ${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3=a^2\frac{d^{2}y}{d{x}^2}$
• B. ${\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
• C. ${\left[1+\left(\frac{dy}{dx}\right)\right]}^3=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
• D. None of the above

Answer 1

Answer: (B)

Given, $(x-h{)}^2+(y-k{)}^2=a^2$...(i)
$\Rightarrow 2(x-h)+2(y-k)\frac{dy}{dx}=0$
$\Rightarrow (x-h)+(y-k)\frac{dy}{dx}=0$...(ii)
Again differentiating
$(y-k)=-\frac{1+{\left(\frac{dy}{dx}\right)}^2}{d^{2}y/d{x}^2}$
Putting in Eq. (ii), we get
$x-h=-(y-k)\frac{dy}{dx}$
$=\frac{\left[1+{\left(\frac{dy}{dx}\right)}^2\right]\frac{dy}{dx}}{\frac{d^{2}y}{d{x}^2}}$
Putting in Eq. (i), we get
$\frac{{\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^2{\left(\frac{dy}{dx}\right)}^2}{{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}$
$+\frac{{\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^2}{{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}=a^2$
$\Rightarrow {\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^2\left[{\left(\frac{dy}{dx}\right)}^2+1\right]=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$
$\Rightarrow {\left[1+{\left(\frac{dy}{dx}\right)}^2\right]}^3=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$