Previous Year Papers

Download Solved Question Papers Free for Offline Practice and view Solutions Online.

Test Series

Take Zigya Full and Sectional Test Series. Time it out for real assessment and get your results instantly.

Test Yourself

Practice and master your preparation for a specific topic or chapter. Check you scores at the end of the test.

Differential Equations

Short Answer Type

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

124 Views

Long Answer Type

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

Let r be the radius of the circle whose centre is C and CM ⊥ x-axis, CN ⊥ y-axis.
∵ circle touches both the axes
∴ CM = CN = r
∴ C is (r, r)
∴ equation of circle is
(x - r)² + (y – r)² = r²    ...(1)
Differentiating both sides w.r.t.x,
$2 left parenthesis straight x minus straight r right parenthesis plus 2 left parenthesis straight y minus straight r right parenthesis space dy over dx space equals space 0$
or $open parentheses straight x minus straight r close parentheses plus left parenthesis straight y minus straight r right parenthesis space straight y subscript 1 space equals space 0$
or $straight x plus straight y space straight y subscript 1 space equals space left parenthesis 1 plus straight y subscript 1 right parenthesis straight r space space space rightwards double arrow space space space space straight r space equals space fraction numerator straight x plus space straight y space straight y subscript 1 over denominator 1 plus straight y subscript 1 end fraction$
Putting this value of r in (1), we get
   $open parentheses straight x minus fraction numerator straight x plus space straight y space straight y subscript 1 over denominator 1 plus space straight y subscript 1 end fraction close parentheses squared plus space open parentheses straight y minus fraction numerator straight x space plus straight y space straight y subscript 1 over denominator 1 plus straight y subscript 1 end fraction close parentheses squared space equals space open parentheses fraction numerator straight x plus straight y space straight y subscript 1 over denominator 1 plus space straight y subscript 1 end fraction close parentheses squared$
or    $left parenthesis straight x plus space straight x space straight y subscript 1 space minus straight x minus space straight y space straight y subscript 1 right parenthesis squared plus space left parenthesis straight y space plus space straight y space straight y subscript 1 minus straight x minus space straight y space straight y subscript 1 right parenthesis squared space equals space left parenthesis straight x space plus space straight y space straight y subscript 1 right parenthesis squared$
or $open square brackets left parenthesis straight x minus straight y right parenthesis space straight y subscript 1 close square brackets squared plus left parenthesis straight y minus straight x right parenthesis squared space equals space left parenthesis straight x plus straight y space straight y subscript 1 right parenthesis squared$
or    $open parentheses straight x minus straight y close parentheses squared plus left parenthesis 1 plus straight y subscript 1 squared right parenthesis space equals space left parenthesis straight x plus straight y space straight y subscript 1 right parenthesis squared$
which is the required differential equation.

86 Views

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

158 Views

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

97 Views

Short Answer Type

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

96 Views

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

135 Views

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.

116 Views

Short Answer Type

88. Form the differential equation representing the family of ellipses having foci on x-axis and centre at the origin.

101 Views

Long Answer Type

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

124 Views

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

158 Views