Transform to axes inclined at 45° to the original axes the
equation x4 + y4 + 6x2y2 = 2.
Answers
Answer:
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Answer:
Let a point with respect to old axis be (x,y)
Let the same point with respect to new axis be (x
′
,y
′
)
If the new axis is formed by rotating old axis by an angle θ then,
x=x
′
cosθ−y
′
sinθ
y=x
′
sinθ+y
′
cosθ
So, here θ=45
∘
cos45
∘
=
2
1
and sin45
∘
=
2
1
So, x=
2
x
′
−
2
y
′
and, y=
2
x
′
+
2
y
′
(1) Putting the value of x and y in x
2
−y
2
=a
2
, we get
(
2
x
′2
+
2
y
′2
−x
′
y
′
)−(
2
x
′2
+
2
y
′2
+x
′
y
′
)=a
2
or, 2x
′
y
′
+a
2
=0
(2) Putting the value of x and y in 17x
2
−16xy+17y
2
=225, we get
17(
2
x
′2
+
2
y
′2
−x
′
y
′
)−16((
2
x
′
−
2
y
′
)(
2
x
′
+
2
y
′
)+17(
2
x
′2
+
2
y
′2
+x
′
y
′
))=225
or, 17(x
′2
+y
′2
)−16(
2
x
′2
−
2
y
′2
)=225
or, 9x
′2
+25y
′2
=225
(3)y
4
+x
4
+6x
2
y
2
=2 can be written as,
(x
2
−y
2
)
2
+8x
2
y
2
=2
Putting the value of x and y in (x
2
−y
2
)
2
+8x
2
y
2
=2,we get
((
2
x
′2
+
2
y
′2
−x
′
y
′
)−(
2
x
′2
+
2
y
′2
+x
′
y
′
))
2
+8(
2
x
′2
+
2
y
′2
−x
′
y
′
)(
2
x
′2
+
2
y
′2
+x
′
y
′
)=2
or, (−2x
′
y
′
)
2
+8(
4
x
′4
+
4
y
′4
−
2
x
′2
y
′2
)=2
or, 4x
′2
y
′2
+2x
′4
+2y
′2
−4x
′2
y
′2
=2
or, x
′4
+y
′4
=1