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

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

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The general equation of such ellipse is

straight x squared over straight a squared plus straight y squared over straight b squared space equals space 1 space space left parenthesis straight a comma space space straight b space being space arbitrary space constants right parenthesis

...(1)

Differentiating w.r.t. x successively two times, we get.

fraction numerator 2 straight x over denominator straight a squared end fraction plus fraction numerator 2 straight y space straight y subscript 1 over denominator straight b squared end fraction space equals space 0 space space space space or space space straight x over straight a squared plus fraction numerator straight y space straight y subscript 1 over denominator straight b squared end fraction space equals space 0

...(2)

rightwards double arrow space space space space straight a squared over straight b squared space equals space minus fraction numerator straight x over denominator straight y space straight y subscript 1 end fraction

...(3)
and

1 over straight a squared plus fraction numerator straight y space straight y subscript 2 plus space straight y subscript 1 squared over denominator straight b squared end fraction space equals space 0 space space space space space space space space rightwards double arrow space space space space straight a squared over straight b squared space equals space fraction numerator negative 1 over denominator straight y space straight y subscript 2 space plus space straight y subscript 1 squared end fraction

...(4)
From (3), (4), we get

fraction numerator negative straight x over denominator straight y space straight y subscript 1 end fraction space equals space fraction numerator negative 1 over denominator straight y space straight y subscript 2 space plus space straight y subscript 1 squared end fraction space space space space space space rightwards double arrow space space space space space straight x left parenthesis straight y space straight y subscript 2 plus straight y subscript 1 squared right parenthesis space minus space straight y space straight y subscript 1 space equals space 0

which is the required differential equation.

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89. Form the differential equation of the family of ellipses having foci on y-axis and centre at origin.

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90. Form the differential equation of the family of hyperbolas having foci on x-axis and centre at origin.

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