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Question

Differential equation of the family of parabolas whose vertex lie on the x axis and focus as origin is

A
y(dydx)2+2xdydxy=0
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B
y(dydx)2+2xdydx+y=0
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C
y(dydx)2+xdydxy=0
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D
y(dydx)2+xdydx+y=0
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Solution

The correct option is A y(dydx)2+2xdydxy=0
focus and vertex both lie on the xaxis
axis of the parabola will be xaxis
Let coordinates of the vertex is (a,0)
Equation of the Directrix will be x+2a=0
Let any point on the parabola be P(x,y)
Now PS=PM
PS2=PM2x2+y2=(x+2a)2x2+y2=x2+4a2+4axy2=4a(a+x)
Differentiating with respect to x
2ydydx=4aa=y2dydxy2=2ydydx(y2dydx+x)y(dydx)2+2xdydxy=0


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