SME
Grade 12 Math · Problem type
Hyperbola: foci and asymptotes
Use this for “Find the foci and asymptotes of x²/9 − y²/16 = 1.”
Diagram
x²/9 − y²/16 = 1
Center (0, 0)
c
=
√
3
2
+
4
2
=
5
asymptotes of slope ±4/3.
This is an example. Type your own values to start a new problem.
(x
−
)²
a
2
−
(y
−
)²
b
2
=
1
c
Center to each focus
worked out (not typed)
s
Asymptote slope (±)
worked out (not typed)
Formulas
c
=
√
a
2
+
b
2
5
=
√
3
2
+
4
2
s
=
b
a
4
3
=
4
3
Assumptions
•
The x term is positive, so the hyperbola opens left and right; the vertices are a from the center.
•
The foci are c from the center, with c² = a² + b².
•
The asymptotes pass through the center with slopes ±b/a; the branches get closer and closer to them.
Step-by-step
WE KNOW
h = 0, k = 0, a = 3, b = 4
FIND
center to each focus (c), asymptote slope (±) (s)
Step 1 · Find center to each focus (c)
Use c = √(a² + b²)
For
a
hyperbola,
c
2
is
the
sum
of
the
squares.
c
=
√
3
2
+
4
2
c
=
√
9 + 16
c
=
√
25
c
=
5
→
foci
(−5,
0)
and
(5,
0);
vertices
(−3,
0)
and
(3,
0)
Step 2 · Find asymptote slope (±): s
Use s = b ÷ a
The
asymptotes
are
the
diagonals
of
the
a
-by-
b
box:
rise
b
over
run
a
.
s
=
4
3
s
=
4
3
→
asymptotes
y
=
±(
4
3
)x
Check
Put the numbers back in. Both sides should match.
5
=
√
3
2
+
4
2
✓
4
3
=
4
3
✓
Topic check
See what you know: 5 questions on this topic
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