- A(5, 0); 1 : 2
- B(6, 2); 1 : 1
- C(5, 0); 2 : 1
- D(0, 5); 1 : 2
Step-by-Step Solution
On the x-axis the y-coordinate is 0. Let AC : CB = k:1:k+18k+(−4)=0 → 8k=4 → k=21, so AC : CB = 1 : 2. Then x=31⋅9+2⋅3=315=5, giving C(5, 0) — one third of the way from A to B, which fits a 1 : 2 split. Reporting 2 : 1 reverses the two parts; (6, 2) is the midpoint of AB, and it does not lie on the x-axis at all.
Answer: A) (5, 0); 1 : 2
Key idea: Section formula and midpoint
**Section formula (internal)**:
A point P dividing segment AB in ratio m:n(P closer to A means m smaller) has coordinates:
P=(m+nmx2+nx1,m+nmy2+ny1)
Where A=(x₁, y₁), B=(x₂, y₂).
**Midpoint** is the special case m=n:M=(2x1+x2,2y1+y2).
**Example**: Find the point dividing AB in ratio 2:3 where A=(1,5),B=(6,−5).x=52⋅6+3⋅1=515=3.y=52⋅(−5)+3⋅5=55=1.
Point: (3, 1).