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Mathematics
Mathematics
Problem
A curve passes through the point
(
1
,
π
6
)
\left(1, \frac{\pi}{6}\right)
(
1
,
6
π
)
. Let the slope of the curve at each point
(
x
,
y
)
(x, y)
(
x
,
y
)
be
y
x
+
sec
(
y
x
)
\frac{y}{x} + \sec\left(\frac{y}{x}\right)
x
y
+
sec
(
x
y
)
,
x
>
0
x>0
x
>
0
. Then the equation of the curve is (A)
sin
(
y
x
)
=
ln
x
+
1
2
\sin\left(\frac{y}{x}\right) = \ln x + \frac{1}{2}
sin
(
x
y
)
=
ln
x
+
2
1
(B)
csc
(
y
x
)
=
ln
x
+
2
\csc\left(\frac{y}{x}\right) = \ln x + 2
csc
(
x
y
)
=
ln
x
+
2
(C)
sec
(
2
y
x
)
=
ln
x
+
2
\sec\left(\frac{2y}{x}\right) = \ln x + 2
sec
(
x
2
y
)
=
ln
x
+
2
(D)
cos
(
2
y
x
)
=
ln
x
+
1
2
\cos\left(\frac{2y}{x}\right) = \ln x + \frac{1}{2}
cos
(
x
2
y
)
=
ln
x
+
2
1
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