AB is a diameter of a circle whose centre is C(2, −3). If A = (−1, 4), find B.

Class 10Coordinate GeometryGeometry — Coordinates

Step-by-Step Solution

The centre of a circle is the midpoint of every diameter, so

(\frac{- 1 + x}{2}, \frac{4 + y}{2}) = (2, - 3)

. Then

- 1 + x = 4

→

x = 5

, and

4 + y = - 6

→

y = - 10

, giving

B

= (5, −10). Computing

y

2 (- 3) + 4 = - 2

adds A's

y

instead of subtracting it; averaging C and A gives (0.5, 0.5), a point inside the circle rather than the far end of the diameter.

Answer: D) (5, -10)

**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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