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Question

The order and degree of the differential equation of the family of the circles touching the x-axis at the origin, are respectively:

A
1,1
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B
1,2
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C
2,1
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D
2,2
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Solution

The correct option is A 1,1
The system of circles touching x-axis at origin will have centres on y-axis.
Let (0,a) be the centre of a circle.
Then the radius of the circle should be a units, since the circle should touch x-axis at origin.
Equation of a circle with centre at (0,a) and radius a is
(x0)2+(ya)2=a2
x2+y22ay=0
The above equation represents the family of circles touching x-axis at origin. Here 'a' is an arbitrary constant.
In order to find the differential equation of system of circles touching x-axis at origin, eliminate the the arbitrary constant from equation
Differentiating equation with respect to y,
2xdxdy+2y2a=0
dxdy=2(ya)
Order is 1 and degree is also 1.
Hence, option 'A' is correct.

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