Let the length of the normal chord through P on the ellipsoid is equal to 4 PG, where
G is the point at which the normal cuts the xy-plane. Find the surface where the point
P lies.
Answers
Step-by-step explanation:
define r.m.s value of current in ac circuit? give the relation between r.m.s value and peak value of current in a.c circuit
Answer:
Normal at point P(acosθ,bsinθ) is
cosθ
ax
−
sinθ
by
=a
2
−b
2
(i)
It meets axes at Q(
a
(a
2
−b
2
)cosθ
,0)
and R(0,−
b
(a
2
−b
2
)sinθ
)
Let T(h,k) is a midpoint of QR.
Then 2h=
a
(a
2
−b
2
)cosθ
and 2k=−
b
(a
2
−b
2
)sinθ
⇒cos
2
θ+sin
2
θ=
(a
2
−b
2
)
2
4h
2
a
2
+
(a
2
−b
2
)
2
4k
2
b
2
=1
⇒ Locus is
4a
2
(a
2
−b
2
)
2
x
2
+
4b
2
(a
2
−b
2
)
2
y
2
=1 (ii)
which is an ellipse, having eccentricity e
′
, given by
e
′2
=1−
4b
2
(a
2
−b
2
)
2
4a
2
(a
2
−b
2
)
2
=1−
a
2
b
2
=e
2
e
′
=e
Note : In Equation (ii),
4a
2
(a
2
−b
2
)
<
4b
2
(a
2
−b
2
)
. Hence, x-axis is minor axis.