Find the equation of the hyperbola whose asymptotes are x + 2y + 3 = 0, 3x + 4y + 5 = 0 and which passes through the point (1, -1).
Answers
Equation of a hyperbola and its asymptotes differ in constant term only.
∴
equation of the hyperbola with asymptotes
x
+
2
y
+
3
=
0
and
3
x
+
4
y
+
5
=
0
can be represented as
(
x
+
2
y
+
3
)
(
3
x
+
4
y
+
5
)
+
λ
=
0
We know that point
(
1
,
−
1
)
lies on the hyperbola
∴
(
1
+
2
×
(
−
1
)
+
3
)
(
3
×
1
+
4
×
(
−
1
)
+
5
)
+
λ
=
0
⟹
λ
=
−
8
Substituting value of
λ
, equation of the hyperbola is
(
x
+
2
y
+
3
)
(
3
x
+
4
y
+
5
)
−
8
=
0
or
3
x
2
+
8
y
2
+
10
x
y
+
14
x
+
22
y
+
7
=
0
We know that
H
+
C
=
2
A
or
C
=
2
A
−
H
, where H is Equation of Hyperbola, C is its conjugate and A is the asymptotes
Solving the above, we can get the equation of the conjugate as
3
x
2
+
8
y
2
+
10
x
y
+
14
x
+
22
y
+
23
=
0
Step-by-step explanation:
Equation of the asymptotes are :
- x + 2y + 3 = 0
- 3x + 4y + 5 = 0
Equation of the hyperbola can be taken as :
- (x + 2y + 3)(3x + 4y + 5) + k = 0
The hyperbola passes through P(1,-1) :
- (1 - 2 + 3)(3 - 4 + 5) + k = 0
- k = -8
Equation of the hyperbola is (x + 2y + 3)(3x + 4y + 5) - 8 = 0
⇒ 3x² + 6xy + 9x + 4xy + 8y² + 12y + 5x + 10y + 15 - 8 = 0
⇒ 3x² + 10xy + 8y² + 14x + 22y + 7 = 0