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Joint Entrance Examination
Mathematics
Differential Equations
Question 340
Question 340 of 490
Medium
MCQ
The differential equation of all circles passing through the origin and having their centres on the
x
x
x
-axis is :
A
\[{y^2} = {x^2} + 2xy{{dy} \over {dx}}\]
B
\[{y^2} = {x^2} - 2xy{{dy} \over {dx}}\]
C
\[{x^2} = {y^2} + xy{{dy} \over {dx}}\]
D
\[{x^2} = {y^2} + 3xy{{dy} \over {dx}}\]
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Official solution
General equation of circles passing through origin and having their centres on the
x
x
x
-axis is \[{x^2} + {y^2} + 2gx = 0\,\,\,\,....\left( i \right)\] On differentiating
w
.
r
.
t
.
x
,
w.r.t.x,
w
.
r
.
t
.
x
,
we get \[2x + 2y.{{dy} \over {dx}} + 2g = 0\] \[ \Rightarrow g = - \left( {x + y{{dy} \over {dx}}} \right)\] \[\therefore\] equation
(
i
)
(i)
(
i
)
be \[{x^2} + {y^2} + 2\left\{ { - \left( {x + y{{dy} \over {dx}}} \right)} \right\}x = 0\] \[ \Rightarrow {x^2} + {y^2} - 2{x^2} - 2x{{dy} \over {dx}}.y = 0\] \[ \Rightarrow {y^2} = {x^2} + 2xy{{dy} \over {dx}}\]
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