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x + xy + y + ax2, 2ax2 – 2y + 3x; – 5xy2 – ax2 – 2y; – 5x + 3y - 20x2
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Given family of parabolas
y=
3
a
3
x
2
+
2
a
2
x
−2a
⇒
a
3
/3
y
=x
2
+
2⋅a
3
a
2
⋅3
x−
a
3
/3
2a
⇒
a
3
3y
+
a
3
6a
=x
2
+2(
4a
3
)x+
16a
2
9
−
16a
2
9
⇒
a
3
3y
+
a
3
6a
+
16a
2
9
=(x+
4a
3
)
2
⇒
a
3
3y
+
a
2
6
+
16a
2
9
=(x+
4a
3
)
2
⇒(x+
4a
3
)
2
=
a
3
3y
+
16a
2
105
⇒(x+
4a
3
)
2
=
a
3
3
(y+
16
35
a) .....(i)
Let x+
4a
3
=X and y+
16
35
a=Y
∴ equation (i) reduces to
X
2
=
a
3
3
Y⇒ vertex
X=0,Y=0
⇒x+
4a
3
=0⇒x=−
4a
3
y+
16
35
a=0⇒y=−
16
35
a
∴xy=−
4a
3
×(−
16
35
a)=
64
105
∴ required locus is : xy=
64
105
hope it helps u dear.......☺️☺️
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