Question

Find the coordinates of a point on \[y - axis\] which are at a distance of \[5\sqrt 2 \] from the point \[P\left( {3, - 2,5} \right)\] .

Answer 1

Hint: We have to find the value of the coordinate of a point on \[y - axis\] which is at a distance of \[5\sqrt 2 \] from the point \[P\left( {3, - 2,5} \right)\] . We solve this question using the concept of the distance of the coordinates . We should also have the knowledge of the value of the other two axes on the \[y - axis\] . We will put the given values in the distance formula and on further solving the equation , we get the value of the coordinate on the \[y - axis\] which is at a distance of \[5\sqrt 2 \] from the point \[P\left( {3, - 2,5} \right)\] .

Complete step-by-step answer:
Given :
Distance between the two points is \[5\sqrt 2 \] . The other point is on \[y - axis\] .
We know that the value of \[x\] and \[z\] of a point on \[y - axis\] is always \[0\] .
So , let the point on \[y - axis\] be \[Q\left( {0,y,0} \right)\] .
Also , we know that the distance formula for two points \[A\] and \[B\] is given as :
\[distance = \sqrt {{{\left( {x2 - x1} \right)}^2} + {{\left( {y2 - y1} \right)}^2} + {{\left( {z2 - z1} \right)}^2}} \]

Now , using the above formula and putting the values , we get the expression as :
\[5\sqrt 2 = \sqrt {{{\left( {3 - 0} \right)}^2} + {{\left( { - 2 - y} \right)}^2} + {{\left( {5 - 0} \right)}^2}} \]
Squaring both sides and simplifying , we get the expression as :
\[25 \times 2 = {\left( 3 \right)^2} + {\left( { - 2 - y} \right)^2} + {\left( 5 \right)^2}\]
Now on further solving , we get the expression as :
\[50 = 9 + {\left( { - 2 - y} \right)^2} + 25\]
\[16 = {\left( { - 2 - y} \right)^2}\]
Taking square root , we get the expression as :
\[ - 2 - y = \pm 4\]
We get the value of the coordinate as :
\[ - 2 - y = 4\] or \[ - 2 - y = - 4\]
On solving , we get two values as :
\[y = - 6\] or \[y = 2\]
Hence , we get the points on \[y - axis\] as \[\left( {0, - 6,0} \right)\] and \[\left( {0,2,0} \right)\] .

Note: We can take a point as \[x_1\] or \[x_2\] in the distance formula and same for the other terms . If a point is on a particular axis then the value of the other two is always zero .
The coordinates of a point on various is given as :
\[x - axis:\left( {x,0,0} \right)\]
\[y - axis:\left( {0,y,0} \right)\]
\[z - axis:\left( {0,0,z} \right)\]