x = e ^ t , y = sin t t = pi/2
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Class 12
>>Maths
>>Continuity and Differentiability
>>Second Order Derivatives
>>If x = e^t sin t, y = e^t ...
Question
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If x=e
t
sint,y=e
t
cost are parametric equations, then
dx
2
d
2
y
at (1,1) is equal to :
Medium
Solution
verified
Verified by Toppr
Correct option is
A
−
2
1
x=e
t
sint, .... (i) and y=e
t
cost .... (ii) are the parametric equations
At point (1,1), we get
1=e
t
sint and 1=e
t
cost
⇒tant=1⇒t=
4
π
Now,
dt
dy
=e
t
(cost−sint)
and
dt
dx
=e
t
(sint+cost)
Also,
dx
dy
=
dt
dx
dt
dy
=
sint+cost
cost−sint
⇒
dx
2
d
2
y
=
dt
d
(
dx
dy
)
dx
dt
=
dt
d
(
cost+sint
cost−sint
)
dx
dt
=
∣
∣
∣
∣
∣
(cost+sint)
2
(cost+sint)(−sint−cost)−(cost−sint)(−sin+cost)
∣
∣
∣
∣
∣
dx
dt
=
∣
∣
∣
∣
∣
(cost+sint)
2
−(cost+sint)
2
−(cost−sint)
2
∣
∣
∣
∣
∣
dx
dt
=
(cost+sint)
2
−2
.
e
t
(sint+cost)
1
=
(e
t
cost+e
t
sint)
−2
.
(cost+sint)
2
1
=
x+y
−2
.
(cost+sint)
2
1
∴
∣
∣
∣
∣
∣
dx
2
d
2
y
∣
∣
∣
∣
∣
(1,1)
=
1+1
−2
.
(cos
4
π
+sin
4
π
)
2
1
=−
2
1
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