Question

The co-ordinates of the points which are equidistant from the points (-1,5),(1,1) and (9,5) are

• A. (4,1)
• B. (5,4)
• C. (1,5)
• D. (4,5)

Answer 1

Answer:

D. (4,5)

let the points be P(p,q)

$d=\sqrt{(x_2-x_1)^2+(y_2-y_1)^2}$

A=(-1,5)

B=(1,1)

C=(9,5)

$d_{PA}=\sqrt{(p+1)^2+(q-5)^2}$

$d_{PB}=\sqrt{(p-1)^2+(q-1)^2}$

$d_{PC}=\sqrt{(p-9)^2+(q-5)^2}$

$d_{PA}=d_{PB}$

$(p+1)^2+(q-5)^2=(p-1)^2+(q-1)^2$

$p^2+2p+1+q^2-10q+25$

$=p^2-2p+1+q^2-2q+1$

$4p-8q+24=0$

$p-2q+6=0$

$d_{PC}=d_{PB}$

$(p-9)^2+(q-5)^2=(p-1)^2+(q-1)^2$

$p^2-18p+81+q^2-10q+25$

$=p^2-2p+1+q^2-2q+1$

$-16p-8q+104=0$

$-2p-q+13=0$

$p-2q+6=0 \times 2$

$2p-4q+12=0$

$-2p-q+13=0$

$-5q=-25$

$q=5$

$p-2q+6=0$

$p-4=0$

$p=4$

(4,5)