If (−1, 2) is the midpoint of the segment joining P(3, p) and Q(q, −4), find p and q.

Class 10Coordinate GeometryGeometry — Coordinates

Step-by-Step Solution

Match the midpoint coordinates:

\frac{3 + q}{2} = - 1

→

3 + q = - 2

→

q = - 5

, and

\frac{p + ( - 4 )}{2} = 2

→

p - 4 = 4

→

p = 8

. Forgetting to multiply the midpoint by 2 (writing

3 + q = - 1

and

p - 4 = 2

) gives the pair

p = 6, q = - 4

; losing the minus sign on the midpoint'

s - 1

gives

q = 5

Answer: B)

p = 8, q = - 5

**Section formula (internal)**:
A point

P

dividing segment

A B

in ratio

m : n (P

closer to A means

m

smaller) has coordinates:

P = (\frac{m x _{2} + n x _{1}}{m + n}, \frac{m y _{2} + n y _{1}}{m + n})

Where

A = (x

₁,

y

₁),

B = (x

₂,

y

₂).

**Midpoint** is the special case

m = n : M = (\frac{x _{1} + x _{2}}{2}, \frac{y _{1} + y _{2}}{2})

.

**Example**: Find the point dividing

A B

in ratio 2:3 where

A = (1, 5), B = (6, - 5) . x = \frac{2 \cdot 6 + 3 \cdot 1}{5} = \frac{15}{5} = 3. y = \frac{2 \cdot ( - 5 ) + 3 \cdot 5}{5} = \frac{5}{5} = 1

.
Point: (3, 1).

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