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Joint Entrance Examination
Mathematics
Differential Equations
Question 336
Question 336 of 490
Medium
MCQ
The differential equation for the family of circle
x
2
+
y
2
−
2
a
y
=
0
,
{x^2} + {y^2} - 2ay = 0,
x
2
+
y
2
−
2
a
y
=
0
,
where a is an arbitrary constant is :
A
\[\left( {{x^2} + {y^2}} \right)y' = 2xy\]
B
\[2\left( {{x^2} + {y^2}} \right)y' = xy\]
C
\[\left( {{x^2} - {y^2}} \right)y' =2 xy\]
D
\[2\left( {{x^2} - {y^2}} \right)y' = xy\]
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Official solution
\[{x^2} + {y^2} - 2ay = 0\,\,\,\,\,\,\,\,\,\,\,...\left( 1 \right)\] Differentiate, \[2x + 2y{{dy} \over {dx}} - 2a{{dy} \over {dx}} = 0\] \[\,\,\,\,\,\,\,\,\,\,\,\,\,\, \Rightarrow a = {{x + yy'} \over {y'}}\] Put in
(
1
)
,
(1),
(
1
)
,
\[{x^2} + {y^2} - 2\left( {{{x + yy'} \over {y'}}} \right) = y = 0\] \[ \Rightarrow \left( {{x^2} + {y^2}} \right)y' - 2xy - 2{y^2}y' = 0\] \[ \Rightarrow \left( {{x^2} - {y^2}} \right)y' = 2xy\]
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