Question

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

• A. ${[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2\frac{d^{2}y}{d{x}^2}$
• B. ${[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2(\frac{d^{2}y}{d{x}^2}{)}^2$
• C. ${[1+\left(\frac{dy}{dx}\right)]}^4=a^2(\frac{d^{2}y}{d{x}^2}{)}^2$
• D. $Noneofthese$

Answer 1

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

$(x-h{)}^2+(y-k{)}^2=a^2$, a is constant ......... (1)

Differentiate (1) w.r.t x

$(x-h)+(y-k)y_1=0$ ........... (2)

Differentiate (2) w.r.t x

$1+(y-k)y_2+y_1^2=0$ .......... (3)

(3) $\Rightarrow y-k=\frac{-(1+y_1^2)}{y_2}$

(2) $\Rightarrow x-h=\frac{(1+y_1^2)y_1}{y_2}$

(1) $\Rightarrow \frac{(1+y_1^2)y_1^2}{y_2^2}+\frac{(1+y_1^2{)}^2}{y_2^2}=a^2$

$\Rightarrow (1+y_1^2{)}^3=a^2{y}_2^2$

(i.e.) ${[1+{\left(\frac{dy}{dx}\right)}^2]}^3=a^2{\left(\frac{d^{2}y}{d{x}^2}\right)}^2$ which is required D.E.