Recomendados
Mais conteúdo relacionado
Semelhante a SUEC 高中 Adv Maths (Locus) (Part 2).pptx
Semelhante a SUEC 高中 Adv Maths (Locus) (Part 2).pptx (20)
Mais de tungwc
Mais de tungwc (20)
Último
Último (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
Notas do Editor
- 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