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

Answer 1

Answer: (B)

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