Question 1165093


Here we have a circle, so the equation will be in {{{(x-x[0])^2 + (y-y[0])^2 = r^2 }}}where ({{{x[0]}}}, {{{y[0]}}}) is the center of the circle.


For part a, the radius {{{r}}} of the circle representing the possible locations of the epicenter is the distance from the station to the epicenter, which is{{{ r=6}}} km.

 
Since given that the station is at ({{{0}}},{{{3}}}), the center of the circle is at that point, and the equation becomes

 {{{(x-0)^2 + (y-3)^2 = 6^2}}}
 
 {{{x^2 + (y-3)^2 = 36}}}
 
This is the equation of the curve that contains the possible location of the epicenter.


For part b, when the epicenter is {{{2km}}} away from the shore, we need to find the {{{x}}}-coordinates where {{{y = 2}}}.

Substitute {{{y = 2}}} into the equation {{{x^2 + (y-3)^2 = 36}}}:

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

{{{x^2 + (-1)^2 = 36}}}

{{{x^2 + 1 = 36}}}

{{{x^2  = 35}}}

{{{x=sqrt(35)}}}

solutions: {{{x=5.92}}} or {{{x=-5.92}}}


so, the coordinates of the epicenter are ({{{5.92}}}, {{{2}}}) or ({{{-5.92}}}, {{{2}}})