More Related Content
Similar to SUEC 高中 Adv Maths (Locus) (Part 2).pptx
Similar to SUEC 高中 Adv Maths (Locus) (Part 2).pptx (20)
More from tungwc
More from tungwc (20)
Recently uploaded
Recently uploaded (20)
SUEC 高中 Adv Maths (Locus) (Part 2).pptx
- 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
Editor's Notes
- https://www.cuemath.com/questions/prove-that-the-perpendicular-at-the-point-of-contact-to-the-tangent-to-a-circle-passes-through-the-centre/
- ntre/
- Knowing the equation of the circle and the coordinates of the tangent point, find the slope of the tangent.
- https://www.onlinemathlearning.com/circle-tangent-angle.html
- From a point P (-5, -10) outside the circle, x^2 + y^2 = 25, 2 tangent lines are drawn to the circle and the tangent points are A & B respectively. Find the equation of the line AB and the length of the chord.
- https://www.onlinemath4all.com/two-circles-are-touching-each-other.html