1. Hyper(Para)bola
𝑎𝑏= 1 2

2. The two curves just touch, as shown.
𝑥 2 − 𝑦 2 = 0.1 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦

3. 𝒂 𝒃 5
1 10 4
1 8
5 2
1 5 2
1 4 1
1 2
1 3
3 2
1 6 3
1 7
7 2
1 12 6
1 14 7 𝒃 𝒂 5
1 10 4
1 8
5 2
1 5 2
1 4 1
1 2
1 3
3 2
1 6 3
1 7
7 2
1 12 6
1 14 7
Answers:
𝑥 2 − 𝑦 2 = 𝑏 2
Find your value of 𝑏 in the table to locate the appropriate value of 𝑎.
Can’t find it?
Try this table 

4. Solve using substitution. 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 𝑏 2 𝑥 2 − 𝑎 2 𝑥 4 = 𝑏 2
𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 𝑏 2
𝑥 2 − 𝑎 2 𝑥 4 = 𝑏 2
𝑥 2 − 𝑎 2 𝑥 = 𝑏 2
𝑎 2 𝑥 − 𝑥 2 + 𝑏 2 =0
𝑥 2 = −1± 1−4 𝑎 2 𝑏 𝑎 2
For a double root (tangent) the discriminant = 0, so
4 𝑎 2 𝑏 2 =1
𝑎𝑏= 1 2
with intersection at ± 1 𝑎 2 ,𝑏
Add comparing gradients method of solution

6. Resources

7. 𝑥 2 − 𝑦 2 = 0.1 2 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 0.2 2 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 0.25 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
𝑥 2 − 𝑦 2 = 0.2 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
SIC_20
SIC_20
𝑥 2 − 𝑦 2 =
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
𝑥 2 − 𝑦 2 = 0.5 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
SIC_20
SIC_20

8. 𝑥 2 − 𝑦 2 = 1 2 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 3 2 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 6 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
𝑥 2 − 𝑦 2 = 3 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
SIC_20
SIC_20
𝑥 2 − 𝑦 2 = 6 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
𝑥 2 − 𝑦 2 = 5 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
SIC_20
SIC_20

9. 𝑥 2 − 𝑦 2 = 7 2 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 3.5 2 𝑦=𝑎 𝑥 2 𝑥 2 − 𝑦 2 = 4 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
𝑥 2 − 𝑦 2 = 3.5 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
SIC_20
SIC_20
𝑥 2 − 𝑦 2 = 4 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
𝑥 2 − 𝑦 2 = 1.5 2
𝑦=𝑎 𝑥 2
The two curves just touch, as shown.
What is the value of 𝑎 ? 𝑥 𝑦
SIC_20
SIC_20