Question

The D. E of the family of parabolas having their focus at the origin and axis along the x-axis is

• A. $y_1[y{y}_1-2x]=y$
• B. $y_1(y_1{)}^2=2x{y}_1+y$
• C. $y{y}_1^2+2x{y}_1=y$
• D. $y{y}_1+2x=y$

Answer 1

Answer: (C)

$y{y}_1^2+2x{y}_1=y$

Given that the equations of the family of parabolas have the focus on origin and axis as X-axis, i.e., as shown the above figure.

As the axis of the family of parabolas is X-axis, the equation of the parabolas can be $y^2=4a(x+k)$, but given that focus is origin.

$\therefore a=k$

$⟹y^2=4a(x+a)$

Here $a$ is the parameter,

$\therefore$ to find the differential equation of the family of curves we need to eliminate the arbitary constant i.e., $a$

Differentiating the equation of the curves with respet to $x$ gives,

$\frac{dy}{dx}=\frac{2a}{y}$

$⟹\frac{y\dot{y}}{2}=a$

Substituting the value of a in equaion of curves gives,

$y^2=y\dot{y}(2x+y\dot{y})$

rearranging the terms gives,

$y{y}_1^2+2x{y}_1=y$