A straight canal runs from a pump house at P(−5, −2) to a field gate at Q(9, 12), the units being metres. A control valve is fitted at the point dividing PQ internally in the ratio 5 : 2. Where is the valve?
Class 10Coordinate GeometryGeometry — Coordinates
A(5, 8)
B(-1, 2)
C(2, 5)
D(8, 5)
Step-by-Step Solution
x=5+25⋅9+2⋅(−5)=745−10=5; y=75⋅12+2⋅(−2)=760−4=8. The valve is at (5, 8) — closer to the gate, as a 5 : 2 split should be. Swapping the ratio to 2 : 5 gives (−1, 2), which sits closer to the pump house instead; (2, 5) is the midpoint, which would be a 1 : 1 split; (8, 5) has the coordinates swapped.
Answer: A) (5, 8)
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).
Master this chapter in 30 minutes a day.
Don't just memorise the answer — build a daily practice habit.