Question

If the distance of the point (4, a) from x-axis is half its distance from y-axis, then ___________.

Answer 1

The distance between $\left(x_1,y_1\right)\mathrm{and}\left(x_2,y_2\right)\mathrm{is}\sqrt{{\left(x_2-x_1\right)}^2+{\left(y_2-y_1\right)}^2}$.

The distance between the point (4, a) and x-axis can be determined by assuming a point (4, 0) on x-axis.

Therefore, the distance between (4, a) and (4, 0)$=\sqrt{{\left(4-4\right)}^2+{\left(0-a\right)}^2}$
$$\begin{gathered} =\sqrt{{\left(a\right)}^2} \\ =a \end{gathered}$$

Also, the distance between the point (4, a) and y-axis can be determined by assuming a point (0, a) on y-axis.

Therefore, the distance between (4, a) and (0, a)$=\sqrt{{\left(4-0\right)}^2+{\left(a-a\right)}^2}$
$$\begin{gathered} =\sqrt{{\left(4\right)}^2} \\ =4 \end{gathered}$$

According to the question,
Distance of the point (4, a) from x-axis is half its distance from y-axis.
$$\begin{gathered} \Rightarrow a=\frac{1}{2}\left(4\right) \\ \Rightarrow a=2 \end{gathered}$$

Hence, if the distance of the point (4, a) from x-axis is half its distance from y-axis, then the value of a is 2.