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 :
$x - axis:\left( {x,0,0} \right)$
$y - axis:\left( {0,y,0} \right)$
$z - axis:\left( {0,0,z} \right)$