Question

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

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

Answer 1

Answer: (B)

${[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

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
$(y-k)=-\frac{1+{\left(\frac{dy}{dx}\right)}^2}{\frac{d^{2}y}{d{x}^2}}$

Putting in Eq. (ii), we get
$x-h=-(y-k)\frac{dy}{dx}$

$=\frac{[1+{\left(\frac{dy}{dx}\right)}^2]\frac{dy}{dx}}{\frac{d^{2}y}{d{x}^2}}$

Putting in Eq. (i), we get
$=\frac{{[1+{\left(\frac{dy}{dx}\right)}^2]}^2{\left(\frac{dy}{dx}\right)}^2}{{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}+\frac{{[1+{\left(\frac{dy}{dx}\right)}^2]}^2}{{\left(\frac{d^{2}y}{d{x}^2}\right)}^2}=a^2$

$\Rightarrow {[1+{\left(\frac{dy}{dx}\right)}^2]}^2[{\left(\frac{dy}{dx}\right)}^2+1]=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$

$\Rightarrow {[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$