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Byju's Answer
Standard XII
Mathematics
Slope Form of Normal : Hyperbola
The equation ...
Question
The equation of the directrix of a hyperbola is x − y + 3 = 0. Its focus is (−1, 1) and eccentricity 3. Find the equation of the hyperbola.
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Solution
Let S be the focus and
P
x
,
y
be any point on the hyperbola.
Draw PM perpendicular to the directrix.
By definition:
SP = ePM
⇒
x
-
-
1
2
+
y
-
1
2
=
3
×
x
-
y
+
3
2
Squaring both the sides, we get:
x
+
1
2
+
y
-
1
2
=
9
2
x
-
y
+
3
2
⇒
x
2
+
2
x
+
1
+
y
2
-
2
y
+
1
=
9
2
x
2
+
y
2
+
9
-
2
x
y
-
6
y
+
6
x
⇒
2
x
2
+
4
x
+
2
+
2
y
2
-
4
y
+
2
=
9
x
2
+
9
y
2
+
81
-
18
x
y
-
54
y
+
54
x
⇒
7
x
2
+
7
y
2
+
50
x
-
50
y
-
18
x
y
+
77
=
0
Equation of the hyperbola:
7
x
2
+
7
y
2
+
50
x
-
50
y
-
18
x
y
+
77
=
0
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Q.
The equation of the directrix of a hyperbola is x − y + 3 = 0. Its focus is (−1, 1) and eccentricity 3. Find the equation of the hyperbola.