If a variable place in 3-0 space moves in such a way that the sum of its intercepts on the x and y-axis exceeds, the reciprocal of its intercept on the z-axis is by 2, then all such plane will pass through the point:
a) (1/2, 1/2, -1/2)
b) (1/2, 1/2, 1/2)
c) (1/2, -1/2, -1/2)
d) (1/2, -1/2, 1/2)
Answers
Explanation:
Let the equation of the variable plane be
a
x
+
b
y
+
c
z
=1 ...(1)
The intercepts on the coordinate axes are a,b,c.
The sum of reciprocals of intercepts in constant λ, therefore
a
1
+
b
1
+
c
1
=λ⇒
a
(1/λ)
+
b
(1/λ)
+
c
(1/λ)
=λ
∴(
λ
1
,
λ
1
,
λ
1
) lies on the plane (1)
Hence, the variable plane (1) always passes through the fixed point (
λ
1
,
λ
1
,
λ
1
ᵈᵉˢⁱ ᶜʰᵒʳᵃ
Answer:
Let the equation of the variable plane be
a
x
+
b
y
+
c
z
=1 ...(1)
The intercepts on the coordinate axes are a,b,c.
The sum of reciprocals of intercepts in constant λ, therefore
a
1
+
b
1
+
c
1
=λ⇒
a
(1/λ)
+
b
(1/λ)
+
c
(1/λ)
=λ
∴(
λ
1
,
λ
1
,
λ
1
) lies on the plane (1)
Hence, the variable plane (1) always passes through the fixed point (
λ
1
,
λ
1
,
λ
1
)
Hope this will help you