Question 43222
CONSTRUCTIONAL METHOD:


First locate the point P(3,4) on a graph paper.
Then draw a circle with centre at (3,4) and radius equal to 5 units.
This circle intersects the y-axis at two points A and B.
Find out the coordinates of these points from the graph.
You shall find coordinates of A and B are (0,8) and (0,0) respectively.


{{{drawing(400,400,-10,10,-10,10,
grid(1),
circle(3,4,5),
locate(3,4,P),
locate(0.2,8,A),
locate(0.3,1,B)
)}}}


Both points A and B lies on y-axis and are at a distance of 5 units from P but only A lies on the positive y-axis. So the required point is A and the coordinates are (0,8).




ALTERNATIVE METHOD:


The distance 'd' between two points ({{{x[1]}}},{{{y[1]}}}) and ({{{x[2]}}},{{{y[2]}}}) is given
{{{d = sqrt((x[1]-x[2])^2 + (y[1]-y[2])^2)}}} ____________(1)


In this problem let the coordinates of A and B are ({{{x[1]}}},{{{y[1]}}}) and ({{{x[2]}}},{{{y[2]}}}) respectively.
Here, {{{d}}} = 5, {{{x[1]}}} = 3, {{{y[1]}}} = 4, {{{x[2]}}} = 0 and {{{y[2]}}} = ?
[Note {{{x[2] = 0}}} because the point ({{{x[2]}}},{{{y[2]}}}) has to lie on y-axis] 
Putting these values in (1) we have
{{{5 = sqrt((3-0)^2 + (4-y[2])^2)}}}
Squaring both sides
{{{5^2 = 3^2 + (4-y[2])^2}}}
or {{{5^2 - 3^2 = (4-y[2])^2}}}
or {{{25 - 9 = (4-y[2])^2}}}
or {{{16 = (4-y[2])^2}}}
or {{{4^2 - (4-y[2])^2 = 0}}}
or {{{(4 + (4-y[2]))(4 - (4-y[2])) = 0}}}
or {{{(8-y[2])y[2]) = 0}}}


Hence either {{{8 - y[2]}}} = 0 i.e. {{{y[2]}}} = 8 or {{{y[2]}}} = 0.
As the reqd. point must be on positive y-axis, so {{{y[2] = 8}}} is the only possibility.


Thus the coordinates of the point A which lie on positive y-axis and is at a distance of 5 units from (3,4) are (0,8).