Point A has a distance 2√26 units from B with coordinates (-5,6) and a distance 5√5 units from C (7,1). Find the coordinates of point A.

+1 vote

The coordinates of point A = (x, y) say

The mid point of AC is B.

So, AC midpoint is equal to B

A(x, y) and C(7, 1)

Mid-point formula: [(x₁ + x₂)/2, (y₁ + y₂)/2]

[(x + 7)/2, (y + 1)/2] = [-5, 6]

(x + 7)/2 = -5 and (y + 1)/2 = 6

Take (x + 7)/2 = -5

Multiply each side by 2

x + 7 = -10

Subtract 7 from each side

x = -17

Take (y + 1)/2 = 6

Multiply each side by 2

y + 1 = 12

Subtract 1 from each side

y = 11

The coordinates of point A(x, y) = (-17, 11).

Above answer and diagram is wrong

Let point A = (x, y), B =(-5,6)  and C = (7,1)

Distance of  A , B points = √[(x + 5)2 + (y - 6)2] = 2√26    -(1)

Distance of A , C points = √[(x  - 7)2 + (y - 1)2] = 5√5       -(2)

First equation simplify then

√[(x + 5)2 + (y - 6)2] = 2√26

Take square to each side

(x + 5)2 + (y - 6)2 = [2√26 ]2

x2 + 10x  + 25 + y2 - 12y + 36 = 104

x2 + y2 + 10x - 12y  = 43    -(3)

Second equation simplify then

√[(x  - 7)2 + (y - 1)2] = 5√5

Take square to each side

(x - 7)2 + (y - 1)2 = 125

x2 + y2 - 14x - 2y = 75   -(4)

Subtract (4) from (3)

24x - 10y = -32

12x - 5y = -16

It have one equation . There is no solution.