生成讲解视频---Here is the extracted content from the image:
**Question 21:**
(8 分) (2024 七年级·湖南长沙·期末) 在平面直角坐标系中, 已知点 A(a,0), B(b,3), C(4,0), 且满足 $|a+b| + (a-b+6)^2 = 0$. 线段 AB 交 y 轴于点 F, 点 D 是 y 轴正半轴上的一点.
**Diagram Description:**
* **Figure 1:**
* Type: Geometric figure on a coordinate plane.
* Elements: X-axis and Y-axis intersecting at origin O. Points A, F, B, C are plotted. Line segment AB intersects the Y-axis at F. Line segment AC is on the X-axis. BC is a line segment forming a triangle ABC. Point A is on the negative X-axis, O is the origin, C is on the positive X-axis. Point B is in the first quadrant. Point F is on the positive Y-axis.
* Labels: x, y, O, A, F, B, C, 图 1.
* **Figure 2:**
* Type: Geometric figure on a coordinate plane, similar to Figure 1 but with additional elements.
* Elements: X-axis and Y-axis intersecting at origin O. Points A, F, B, C are plotted. D is a point on the positive Y-axis. M is a point inside triangle ABD. Lines/segments: AB, AC, BC, AF, BF, AD, BD, AM, DM, MN (dashed line). N is a point on the negative X-axis. There are angle bisector markings on ∠CAB and ∠ODB.
* Labels: x, y, O, A, F, D, M, B, C, N, 图 2, α is marked as ∠CAB.
* **Figure 3:**
* Type: Geometric figure on a coordinate plane, similar to Figure 1.
* Elements: X-axis and Y-axis intersecting at origin O. Points A, F, B, C are plotted. Line segment AB intersects the Y-axis at F. Line segment AC is on the X-axis. BC is a line segment forming a triangle ABC. Point A is on the negative X-axis, O is the origin, C is on the positive X-axis. Point B is in the first quadrant. Point F is on the positive Y-axis.
* Labels: x, y, O, A, F, B, C, 图 3.
**Question 21 Sub-questions:**
(1) 求出点 A, B 的坐标;
(2) 如图 2, 若 DB∥AC, ∠BAC = α, AM, DM 分别平分∠CAB, ∠ODB, 求∠AMD (用含 α 的代数式表示);
(3) 如图 3, 坐标轴上是否存在一点 P, 使得 △ABP 的面积和 △ABC 的面积相等? 若存在请求出 P 点坐标; 若不存在, 请说明理由.
**Question 22:**
(8 分) (2024 七年级·四川凉山·期末) 某体育用品店准备购进甲、乙两种品牌跳绳, 若购买甲种跳绳 10 根, 乙种跳绳 5 根, 需要 100 元, 若购买甲种跳绳 5 根, 乙种跳绳 3 根, 需要 55 元.
**Other Relevant Text:**
* 第 5 页 共 6 页 (Page 5 of 6)
**Handwritten Notes (Likely scratch work):**
* |a+b| + (a-b+6)^2 = 0
* { a+b = 0, a-b+6 = 0 }
* { a = -b, -b-b+6 = 0 }
* { a = -b, -2b = -6 }
* { a = -b, b = 3 }
* { a = -3, b = 3 }
* ∴ A (-3, 0)
* B (3, 3)
* C (4, 0)
* a+b=0 (a+b)2=0
* ∴ a+b=0 a+b=0
* (2)
* ∠BAM = ∠CAM = 1/2∠CAB
* BD || AC
* ∴ ∠BDO + ∠DOA = 180 (likely a mistake in angle notation)
* ∴ BD || AC ∴ ∠BDP + ∠DPC = 180 (likely a mistake in angle notation)
* ∠ODB
* DM 平分 ∠ODB
* ∠ODM = 1/2 ∠ODB
* ∠BDP = 90 (possibly BD ⊥ y-axis, but BD || AC || x-axis implies BD is horizontal, not vertical, maybe related to some other line)
* ∠BDF + ∠FDO = ∠BDO
* ∠BDM = ∠FMD ? ± ∠BDF ? 45° (unclear)
* ∠BAM = ∠CAM = 1/2 α
* ∠AMD = ?
* ∠ADM = ?
* ∠LDMA = ∠DMNT?N?? (unclear)
* = 45° + 1/2 α B (likely an expression for ∠AMD)
* 45° + 1/2 α (repetition)
* BP⊥y轴 (BP perpendicular to y-axis)
* (3)
* 存在点 P (Point P exists)
* △ABP 的面积 = △ABC 的面积