Question 1176758

If({{{x}}},{{{4}}}) is equidistant to ({{{5}}},{{{-2}}}) and ({{{3}}},{{{4}}}) 

means distance between ({{{x}}},{{{4}}}) and ({{{5}}},{{{-2}}}) is same as distance  between ({{{x}}},{{{4}}}) and ({{{3}}},{{{4}}}) 


to find the {{{x}}}, use distance formula


distance between ({{{x}}},{{{4}}}) and ({{{5}}},{{{-2}}}) is:

{{{d=sqrt((x-5)^2+(4-(-2))^2)}}}
{{{d=sqrt((x-5)^2+(4+2)^2)}}}
{{{d=sqrt((x-5)^2+36)}}}......eq.1

distance between ({{{x}}},{{{4}}}) and ({{{3}}},{{{4}}}) is:

{{{d=sqrt((x-3)^2+(4-4)^2)}}}
{{{d=sqrt((x-3)^2+(0)^2)}}}
{{{d=sqrt((x-3)^2)}}}......eq.2

from eq.1 and eq.2 we have

{{{sqrt((x-5)^2+36)=sqrt((x-3)^2)}}}....square both sides

{{{(x-5)^2+36=(x-3)^2}}}.....expend

{{{x^2-10x+25+36=x^2-6x+9}}}

{{{cross(x^2)-10x+61=cross(x^2)-6x+9}}}

{{{-10x+61=-6x+9}}}

{{{-9+61=10x-6x}}}

{{{52=4x}}}

{{{x=52/4}}}

{{{x=13}}}


so, the point is ({{{13}}},{{{4}}})