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Differential Equations

Short Answer Type

81. Form the differential equation of the family of circles having centre on x-axis and radius 3 units.

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Long Answer Type

82. Find the differential equation of all the circles in the first quadrant which touch the co-ordinate axes.

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83. Form the differential equation of the family of circles in the second quadrant and touching the coordinate axes.

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84. Form the differential equation of the family of circles touching the x-axis at origin.

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Short Answer Type

85. Form the differential equation of the family of circles touching the y-axis at origin.

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86. Form the differential equation of the family of parabolas having vertex at origin and axis along positive direction of x-axis.

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Long Answer Type

87. Form the differential equation of the family of parabolas having vertex at origin and axis along positive direction of y-axis.

Let P denote the family of parabolas and let S (0. a) be the focus of a member of the given family, where a is an arbitrary constant.
∴ equation of family P is
a² = 4 a y ...(1)
Differentiating both sides w.r.t. x, we get.
$2 straight x space equals space 4 straight a dy over dx$
$therefore space space space space 4 straight a space equals space fraction numerator 2 straight x over denominator begin display style dy over dx end style end fraction$
Putting value of 4a in (1), we get,
$straight x squared space equals space fraction numerator 2 straight x over denominator begin display style dy over dx end style end fraction straight y space space space or space space space space straight x dy over dx space equals space 2 straight y$
which is the required differential equation.