Question

The D.E of the family of parabolas with vertex at (0, -1) and having axis along the y axis is

• A. $x{y}^1-2y-2=0$
• B. $x{y}^1+y+1=0$
• C. $x{y}^1-y-1=0$
• D. $x{y}^1+2xy+1=0$

Answer 1

The correct option is A $x{y}^1-2y-2=0$

General equation will be

$(x-h{)}^2=4a(y-k)$

as h=0,k=-1

$x^2=4a(y+1)$

Now by differentiating this equation, we get

$2x=4a\frac{dy}{dx}$

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

Using the value of a in equation, we get

$x^2=2x\frac{dx}{dy}(y+1)$

$x^2\frac{dy}{dx}=2xy+2x$

$x^2{y}^1-2xy-2x=0$

$x{y}^1-2y-2=0$