Question

If h, k are the perpendicular distances from (1,2,3) to the x-axis, z-axis respectively. Then hk is equal to:

Answer 1

Hint: The distance of a point from an axis is given by the root of the sum of the squares of the coordinates of the points excluding the coordinate corresponding to the axis from which we are finding the distance. For example: the distance of a point P(a,b,c) from the x-axis is given by $\sqrt{{{b}^{2}}+{{c}^{2}}}$. So, use this to find the values of h and k separately and multiply the results to get the answer.

Complete step-by-step solution:
Let us start the solution to the above question by drawing a representative diagram of the situation given in the question.

The distance of a point from an axis is given by the root of the sum of the squares of the coordinates of the points excluding the coordinate corresponding to the axis from which we are finding the distance.
So, “h” is the distance from the x-axis, so we will add the squares of y-coordinate and z-coordinate of the point and take the root.
$h=\sqrt{{{2}^{2}}+{{3}^{2}}}=\sqrt{4+9}=\sqrt{13}$
Also, k is the distance from the z-axis, so we will add the squares of y-coordinate and x-coordinate of the point and take the root.
$k=\sqrt{{{2}^{2}}+{{1}^{2}}}=\sqrt{4+1}=\sqrt{5}$
Now, let us find the product of h and k, i.e., hk.
$hk=\sqrt{13}\times \sqrt{5}=\sqrt{13\times 5}=\sqrt{65}$
Hence, the answer to the above question is $\sqrt{65}$.

Note: If you are not aware of the point that the distance of a point from an axis is given by the root of the sum of the squares of the coordinates of the points excluding the coordinate corresponding to the axis from which we are finding the distance, you will have to consider a general point on the axis from which you are finding the distance, which will have two coordinates zero and one variable coordinate. Use the distance formula to get the distance in terms of variable coordinate and minimize it to get the perpendicular distance.