(1) The document discusses finding equations of tangent lines to circles and the intersections of those tangent lines. It provides examples of finding the slopes and equations of tangent lines given the circle's center and a point on the circle. (2) Methods are described for finding the angles between two tangent lines to a circle based on their slopes. Examples are given of solving systems of equations to find the points where tangent lines intersect. (3) One example determines the equation of a circle given that it passes through two known points and is tangent to another circle at a third point.

1. 𝒑𝒈 𝟗𝟗
𝑡𝑎𝑛𝑔𝑒𝑛𝑡

2. 𝒑𝒈 𝟏𝟎𝟐

3. 𝒑𝒈 𝟏𝟎𝟎
𝑥1 , 𝑦1
−𝑔 , −𝑓
𝑚𝑂𝑃 =
𝑦1 + 𝑓
𝑥1 + 𝑔
𝑚𝐿𝑃 = −
𝑥1 + 𝑔
𝑦1 + 𝑓
𝐿𝑃 ∶ 𝑦 − 𝑦1 = −
𝑥1 + 𝑔
𝑦1 + 𝑓
(𝑥 − 𝑥1)

4. 𝒑𝒈 𝟏𝟎𝟎
𝑥1 , 𝑦1
𝑡𝑎𝑛𝑔𝑒𝑛𝑡
𝑔 = 𝑓 = 0
𝑥1 , 𝑦1

5. 𝒑𝒈 𝟏𝟎𝟐
𝑥1 , 𝑦1
𝑔 =? , 𝑓 = ?
3𝑥 − 4𝑦
= 25
6𝑥 − 3𝑦 + 5 = 0
𝑥 − 2𝑦 + 13 = 0
2𝑥 = 3𝑦
5𝑥 − 12𝑦 + 17 = 0
𝑦 =
1
3
1
3
2 5
5

6. 𝒑𝒈 𝟏𝟎𝟐
(𝑎) 𝐿𝑒𝑡 𝑂 = 𝑐𝑒𝑛𝑡𝑟𝑒 = (0, 0)
𝐿𝑒𝑡 𝑃 = (−2, 5)
𝑚𝑂𝑃 = −
5
2
𝑚𝐿𝑃 =
2
5
=
2 5
5
𝐿𝑒𝑡 𝐿 = 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒
(𝑎) 𝐿𝑒𝑡 𝑂 = 𝑐𝑒𝑛𝑡𝑟𝑒 = (3, −1)
𝐿𝑒𝑡 𝑃 = (2, 2)
𝑚𝑂𝑃 =
2 + 1
2 − 3
= − 3
𝑚𝐿𝑃 =
1
3
𝐿𝑒𝑡 𝐿 = 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑙𝑖𝑛𝑒

7. 𝒑𝒈 𝟏𝟎𝟏
𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑙 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑎𝑡 𝑦 𝑎𝑥𝑖𝑠, 𝑓𝑖𝑛𝑑 𝑙
0 , 3
0 , −1
𝑔 = 𝑓 = −1

8. 𝒑𝒈 𝟏𝟎𝟏
𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑙 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑡ℎ𝑒 𝑐𝑖𝑟𝑐𝑙𝑒 𝑎𝑡 (−2, 2)
𝑝𝑟𝑜𝑣𝑒 𝑡ℎ𝑎𝑡 𝑡ℎ𝑖𝑠 𝑠𝑡𝑟𝑎𝑖𝑔ℎ𝑡 𝑙𝑖𝑛𝑒 𝑙 𝑖𝑠 𝑡𝑎𝑛𝑔𝑒𝑛𝑡 𝑡𝑜 𝑎𝑛𝑜𝑡ℎ𝑒𝑟 𝑐𝑖𝑟𝑐𝑙𝑒
𝑔 = 0, 𝑓 = 1

9. 𝒑𝒈 𝟏𝟎𝟐
(2, 1), (−2, −1)
25
4
(4, 6)

10. 𝒑𝒈 𝟏𝟎𝟐
𝑘 = 9

11. 𝒑𝒈 𝟏𝟎𝟑

12. 𝒑𝒈 𝟏𝟎𝟑
𝑥1 , 𝑦1
𝑔 = 𝑓 = − 1
𝑥1 , 𝑦1
𝑥 , 𝑦

13. 𝒑𝒈 𝟏𝟎𝟑
(2. 1)
2
5
,
9
5
𝑔 = 𝑓 = − 1
𝑥 = 2
3𝑥 − 4𝑦 + 6 = 0

14. 𝑥 = 2
3𝑥 − 4𝑦 + 6 = 0
𝒑𝒈 𝟏𝟎𝟒
𝑥2 + 𝑦2 − 2𝑥 − 2𝑦 + 1 = 0

15. 𝒑𝒈 𝟏𝟎𝟒 − 𝟏𝟎𝟓
𝑥 = 2
3𝑥 − 4𝑦 + 6 = 0

16. 𝒑𝒈 𝟏𝟎𝟓
𝑓𝑖𝑛𝑑 𝑡ℎ𝑒 2 𝑡𝑎𝑛𝑔𝑒𝑛𝑡𝑠 & 𝑎𝑛𝑔𝑙𝑒 𝑜𝑓 𝑡ℎ𝑒 𝑙𝑖𝑛𝑒𝑠

17. 𝒑𝒈 𝟏𝟎𝟕
24𝑥 + 7𝑦 − 100 = 0
𝑥 − 𝑦 + 10 = 0
3𝑥 − 4𝑦 − 25 = 0
2𝑥 + 𝑦 − 8 = 0
24𝑥 − 7𝑦 − 20 = 0
𝑦 = 4
𝑥 + 𝑦 − 10 = 0
𝑥 = −5
𝑥 − 2𝑦 + 11 = 0
𝑥 = 2
3𝑥 + 4𝑦 = 0 𝑥 = 0
3 𝑥 − 𝑦 + 2 = 0
4𝑥 − 3𝑦 + 2 = 0
3 𝑥 + 𝑦 − 2 = 0
𝑦 =
2
5
tan 𝜃 =
12
5

18. 𝒑𝒈 𝟏𝟎𝟕
3 𝑥 − 𝑦 + 2 = 0
3 𝑥 + 𝑦 − 2 = 0
𝑐𝑒𝑛𝑡𝑟𝑒 = (0, 0) , 𝑟 = 1
𝐿𝑒𝑡 𝐴 = (0, 2)
𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝐴 ∶ 𝑦 − 2 = 𝑚 (𝑥 − 0)
𝑚𝑥 + 𝑦 − 2 = 0
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝐴 𝑡𝑜 𝑐𝑒𝑛𝑡𝑟𝑒 ∶
−2
𝑚2 + 1
= 1
𝑚2 + 1 = 4
𝑚 = ± 3
𝑤ℎ𝑒𝑛 𝑚 = 3, 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝐴 ∶
𝑤ℎ𝑒𝑛 𝑚 = − 3, 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝐴 ∶

19. 𝒑𝒈 𝟏𝟎𝟕
𝑐𝑒𝑛𝑡𝑟𝑒 = (2, 3) , 𝑟 = 2
𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑝𝑎𝑠𝑠𝑖𝑛𝑔 𝑃 ∶ 𝑦 = 𝑚𝑥
𝐿𝑒𝑡 𝑃 = (0, 0)
𝑚𝑥 − 𝑦 = 0
𝑑𝑖𝑠𝑡𝑎𝑛𝑐𝑒 𝑜𝑓 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑎𝑡 𝐴 𝑡𝑜 𝑐𝑒𝑛𝑡𝑟𝑒 (2, 3) ∶
2𝑚 − 3
𝑚2 + 1
= 2
2𝑚 − 3 2 = 4 (𝑚2 + 1)
4𝑚2
+ 9 − 12𝑚 = 4𝑚2
+ 4
12𝑚 = 5
𝑚 =
5
12
𝐴
12
5
𝑚
𝜃
tan 𝜃 =
12
5

20. 𝒑𝒈 𝟏𝟎𝟔
(𝑥. 𝑦)

21. 𝒑𝒈 𝟏𝟎𝟕
65
19
4 5
5
(1. 0) (−3. 4)
𝑥2 + 𝑦2 + 2𝑥 − 4𝑦 − 20
= 0

22. 𝒑𝒈 𝟏𝟎𝟕
𝑘 = 14 𝑘 = 4
𝑘 = 31 𝑘 = −19

23. 𝒑𝒈 𝟏𝟎𝟕
𝑐𝑒𝑛𝑡𝑟𝑒 = (0, 0) , 𝑟 = 5 𝑃 = (𝑥, 𝑦) = (− 5, 10)
−5𝑥1 + 10𝑦1 − 25 = 0
𝑥1 − 2𝑦1 + 5 = 0
𝑥1
2
− 2𝑦1
2
− 25 = 0

24. 𝒑𝒈 𝟏𝟎𝟖
𝑚𝐿 = −2. 𝐹𝑖𝑛𝑑 𝐿

25. 𝒑𝒈 𝟏𝟎𝟖

26. 𝒑𝒈 𝟏𝟎𝟗
2𝑥 − 𝑦 + 10 = 0 2𝑥 − 𝑦 − 10 = 0
4𝑥 + 3𝑦 + 25 = 0 4𝑥 + 3𝑦 − 25 = 0
3𝑥 − 𝑦 + 2 10 = 0 3𝑥 − 𝑦 − 2 10 = 0
𝑏 = ±2 2
3𝑥 − 4𝑦 + 20 = 0 3𝑥 − 4𝑦 − 20 = 0
2𝑥 − 𝑦 + 7 − 4 5 = 0
2𝑥 − 𝑦 + 7 + 4 5 = 0

27. 𝒑𝒈 𝟏𝟎𝟗
3𝑥 + 4𝑦 − 29 = 0
3 5
5
, −
6 5
5
−
3 5
5
,
6 5
5

28. 𝒑𝒈 𝟏𝟏𝟎

29. 𝒑𝒈 𝟏𝟏𝟎

30. 𝒑𝒈 𝟏𝟏𝟏

31. 𝒑𝒈 𝟏𝟏𝟑
13
7𝑥2 + 7𝑦2 − 24𝑥 − 19 = 0

32. 𝒑𝒈 𝟏𝟏𝟑
∴ 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 ∶ 7𝑥2 + 7𝑦2 − 24𝑥 − 19 = 0
𝑔1 = 0, 𝑓1 = 0, 𝑐1 = −1 𝑔1 = 1, 𝑓1 = 0, 𝑐1 = 0
𝑡ℎ𝑒 𝑙𝑖𝑛𝑒 𝑐𝑜𝑛𝑛𝑒𝑐𝑡𝑖𝑛𝑔 𝑏𝑙𝑢𝑒 & 𝑟𝑒𝑑 𝑐𝑖𝑟𝑐𝑙𝑒, 𝑦 = 0
𝐿𝑒𝑡 𝑐𝑒𝑛𝑡𝑟𝑒 𝑜𝑓 𝑔𝑟𝑒𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒, 𝑂 = (𝑥, 0)
𝑟 𝑓𝑜𝑟 𝑔𝑟𝑒𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒 = 𝑂𝐴 = 𝑂𝐵
𝐿𝑒𝑡 𝐵 = (3, 2)
Let 𝐴 = −
1
2
,
3
2
𝑥 +
1
2
2
+ 0 −
3
2
2
= 𝑥 + 3 2
+ 0 − 2 2
𝑥 =
12
7
, 𝑟 =
277
7

33. 𝒑𝒈 𝟏𝟏𝟏 − 𝟏𝟏𝟐

34. 𝒑𝒈 𝟏𝟏𝟐
𝐹𝑖𝑛𝑑 𝑡ℎ𝑒 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛 𝑡ℎ𝑎𝑡 𝑝𝑎𝑠𝑠𝑒𝑠 𝑡ℎ𝑟𝑜𝑢𝑔ℎ 𝑜𝑟𝑖𝑔𝑖𝑛 & 𝑖𝑛𝑡𝑒𝑟𝑠𝑒𝑐𝑡 𝑡ℎ𝑒 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙 𝑐𝑖𝑟𝑐𝑙𝑒𝑠.
𝑔1 = −4, 𝑓1 = 1, 𝑐1 = 6 𝑔1 = −
1
2
, 𝑓1 = 2, 𝑐1 = −3

35. 𝒑𝒈 𝟏𝟏𝟑
𝐴 𝑐𝑖𝑐𝑙𝑒 𝑖𝑠 𝑜𝑟𝑡ℎ𝑜𝑔𝑜𝑛𝑎𝑙 𝑡𝑜 2 𝑐𝑖𝑟𝑐𝑙𝑒𝑠. 𝐹𝑖𝑛𝑑 𝑒𝑞𝑢𝑎𝑡𝑖𝑜𝑛.
𝑔1 = 3, 𝑓1 = 5, 𝑐1 = 12 𝑔1 = 2, 𝑓1 = 1, 𝑐1 = −2
𝑥2
+ 𝑦2
+ 𝑥 + 4𝑦 − 7 = 0 (𝑔𝑟𝑒𝑒𝑛 𝑐𝑖𝑟𝑐𝑙𝑒)

36. 𝒑𝒈 𝟏𝟏𝟑
𝑔1 = ? , 𝑓1 = ? , 𝑐1 = ?
4𝑥2
+ 4𝑦2
+ 18𝑥 − 8𝑦 − 5 = 0

37. 𝒑𝒈 𝟏𝟏𝟑
𝑔1 = ? , 𝑓1 = ? , 𝑐1 = ?
2𝑥2 + 2𝑦2 + 8𝑥 − 13𝑦 + 11 = 0
𝑥2
+ 𝑦2
− 4𝑥 − 8 = 0